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Perpendicular

In geometry, perpendicular describes the relationship between two lines, rays, segments, or planes that intersect at a right angle, measuring exactly 90 degrees (or π/2 radians).^[1] This intersection forms four congruent right angles at the point of meeting, often denoted by a small square symbol (⊥) to indicate the 90-degree orientation.^[2] The concept is fundamental to Euclidean geometry and extends to higher dimensions, such as perpendicular vectors in linear algebra, where their dot product equals zero.^[3] The term "perpendicular" derives from the Latin perpendiculum, meaning a plumb line or plumb bob, an ancient tool used to establish vertical alignment relative to the horizontal plane, reflecting early practical applications in construction and surveying.^[4] Formalized in ancient Greek mathematics, particularly in Euclid's Elements around 300 BCE, the idea of right angles underpinned theorems on triangles, circles, and polygons, enabling precise constructions like the perpendicular bisector.^[5] Key properties include: perpendicular lines always intersect (unlike parallel lines); the product of their slopes in the Cartesian plane is -1 (provided neither is vertical); and if two distinct lines are perpendicular to the same third line, they are parallel to each other.^[6] These attributes ensure perpendicularity's role in proving geometric theorems and solving real-world problems. Beyond pure mathematics, perpendicular relationships appear in physics (e.g., normal forces acting perpendicular to surfaces), engineering (e.g., right-angle frameworks in architecture), and computer graphics (e.g., orthogonal projections).^[7] In three-dimensional space, planes or lines can be perpendicular if their normal vectors are perpendicular, facilitating calculations in fields like crystallography and robotics.^[8] The concept's enduring utility stems from its simplicity and universality across disciplines.

Definition and Fundamentals

Definition

In geometry, two lines are perpendicular if they intersect at a right angle, which measures exactly 90 degrees or \pi/2 radians.^[9]^[10] This relationship is often denoted by the symbol \perp.^[9] This concept extends to other geometric figures: two line segments are perpendicular if they form a right angle at their intersection point, while two rays are perpendicular if the angle between them at their common endpoint is a right angle; similarly, their union must produce a right angle to establish perpendicularity.^[11] For planes, two planes are perpendicular when lines drawn in one plane, perpendicular to the line of intersection of the two planes, are also perpendicular to the second plane.^[12] The term "perpendicular" originates from the Latin perpendiculum, meaning a plumb line, referring to a line hanging vertically by gravity, and entered English in the late 15th century to describe lines at right angles to a horizontal.^[13] Its early mathematical use appears in Euclidean geometry around 300 BCE, where Euclid defined right angles and perpendiculars as foundational elements in Elements.^[10] A right angle of 90 degrees satisfies the trigonometric identities \sin(90^\circ) = 1 and \cos(90^\circ) = 0, which arise from the unit circle where the sine is the y-coordinate and cosine is the x-coordinate at that angle.^[14]

Basic Properties

When two lines intersect at a right angle, they are perpendicular, forming four congruent adjacent angles, each measuring 90 degrees, while the vertically opposite angles are equal due to the properties of intersecting lines.^[15]^[16] This configuration arises from the definition of perpendicularity, ensuring that the angles adjacent to the intersection are complementary to form straight lines, with each pair of opposite angles congruent.^[17] In triangle geometry, an altitude is the perpendicular segment drawn from a vertex to the line containing the opposite side, thereby creating right angles at the foot of the altitude.^[18] This property holds even if the foot lies outside the side in obtuse triangles, emphasizing the perpendicular's role in defining heights and areas.^[19] The perpendicular bisector of a line segment is a line that intersects the segment at its midpoint and at a 90-degree angle, dividing it into two congruent halves.^[20] This symmetry property is fundamental in constructions and proofs involving equal distances from points on the bisector to the segment's endpoints.^[21] In the coordinate plane, for non-vertical lines with slopes m_1 and m_2, two lines are perpendicular if the product of their slopes satisfies m_1 \cdot m_2 = -1.^[22] This condition derives from the fact that the angles formed by the lines must be right angles, leading to the negative reciprocal relationship between slopes.^[23]

Construction Techniques

Geometric Constructions

Geometric constructions of perpendicular lines rely on the classical tools of compass and straightedge, as outlined in Euclid's Elements, enabling the creation of right angles without measurement. These methods form the foundation of Euclidean geometry and demonstrate the constructibility of perpendiculars from basic postulates.^[24] One fundamental construction erects a perpendicular to a given line at a specified point on that line, as described in Euclid's Proposition I.11. To perform this, select an arbitrary point D on the line beyond the given point C. Then, using a compass, mark point E on the opposite side of C such that CE = CD. Next, on segment DE, construct an equilateral triangle FDE. Finally, draw the straight line from C to F, which forms the perpendicular at C. This method ensures the angles at C are right angles through the congruence of triangles DCF and ECF by SSS (side-side-side), making the adjacent angles equal and thus each 90 degrees.^[24] To construct a perpendicular from a point not on the line, Euclid's Proposition I.12 provides a systematic approach using circle intersections. Given line AB and external point C, first choose an arbitrary point H on the opposite side of AB from C. Draw a circle centered at C with radius CH, intersecting AB at points D and E. Then, construct the perpendicular bisector of DE (using methods from prior propositions) to find its midpoint F. The line CF is perpendicular to AB, as F is the midpoint and the construction ensures equal angles through symmetric properties and the definition of perpendicularity.^[25] The perpendicular bisector of a line segment locates the set of points equidistant from the segment's endpoints and intersects the segment at a right angle. For segment AB, open the compass to a radius greater than half AB and place the point at A to draw an arc; repeat from B to draw another arc, intersecting at two points C and D above and below AB. Connect C and D with a straightedge to form the perpendicular bisector, which passes through the midpoint of AB and forms 90-degree angles with it due to the congruent triangles created by the equal arcs.^[26] Thales' theorem offers a direct method to construct a right angle by leveraging circle properties: if A and B are endpoints of a circle's diameter, then any point C on the circle forms angle ACB as 90 degrees. To construct this, draw segment AB as the diameter, center the compass at the midpoint of AB, and draw the circle; select C on the circumference and connect A, C, B to obtain the perpendicular at C relative to AB. This application highlights the role of circles in perpendicular constructions.^[27]

Coordinate-Based Methods

In coordinate geometry, the equation of a line perpendicular to a given line can be determined using slopes or vector methods. For a line given in slope-intercept form y = mx + c, where m is the slope, the slope of any perpendicular line is the negative reciprocal, -1/m. Thus, the equation of the perpendicular line is y = (-1/m)x + k, where k is a constant determined by a point through which the line passes.^[28] To find the foot of the perpendicular from a point (x_0, y_0) to the line ax + by + c = 0, the coordinates (h, k) of the foot satisfy both the line equation and the condition that the line segment from (x_0, y_0) to (h, k) is perpendicular to the given line. The slope of the given line is -a/b, so the slope of the perpendicular is b/a. The equation of the perpendicular line passing through (x_0, y_0) is y - y_0 = (b/a)(x - x_0). Solving this simultaneously with ax + by + c = 0 yields the foot coordinates:

h = \frac{b(bp - aq) - ac}{a^2 + b^2}, \quad k = \frac{a(-bp + aq) - bc}{a^2 + b^2},

where p = x_0 and q = y_0.^[29] Parametric equations provide another algebraic approach for constructing perpendiculars. For the line ax + by + c = 0, the normal vector (a, b) is perpendicular to the line, as the line equation can be rewritten as the dot product (a, b) \cdot (x, y) = -c.^[30] Thus, a line perpendicular to the given line has direction vector (a, b). The parametric equations for the perpendicular from (x_0, y_0) are x = x_0 + at, y = y_0 + bt, where t = -(ax_0 + by_0 + c)/(a^2 + b^2). The foot is then at t along this parametric path. Note that a direction vector for the original line is (-b, a), which is perpendicular to (a, b).^[31] Example: Consider the line $2x + y - 3 = 0 and the point (0, 0). Here a=2, b=1, c=-3. The parametric equations are x=0+2t, y=0+1t. Substitute into the line: $2(2t) + t - 3 = 0, so $5t=3, t=3/5. The foot is at x=6/5, y=3/5. To verify, the slope of the line is -2, so the perpendicular slope is $1/2. The equation is y=(1/2)x. Intersect with y=3-2x: (1/2)x = 3-2x, (5/2)x=3, x=6/5, y=3/5. This matches.

Perpendicularity in Plane Geometry

Relation to Parallel Lines

In Euclidean plane geometry, a fundamental theorem states that if a straight line is perpendicular to one of two parallel straight lines, then it is also perpendicular to the other.^[32] This result, known as the perpendicular transversal theorem, arises when a transversal intersects two parallel lines and forms a right angle with one of them.^[33] When a transversal is perpendicular to one of two parallel lines, it necessarily creates right angles—measuring 90 degrees—with both lines.^[32] This property holds because the angles formed at the intersections are corresponding or alternate interior angles, which are equal due to the parallelism of the lines.^[34] Consequently, if one such angle is a right angle, the corresponding angle with the second parallel line must also be 90 degrees, ensuring perpendicularity.^[32] The proof relies on the properties of alternate interior angles and Euclid's parallel postulate. Consider two parallel lines l_1 and l_2, intersected by a transversal t such that t is perpendicular to l_1, forming a right angle at their intersection. The alternate interior angle formed by t and l_2 (on the opposite side of the transversal from the right angle with l_1) must then be equal to that right angle. By Euclid's Proposition I.29, a straight line falling on parallel straight lines makes the alternate interior angles equal to one another, a result derived using the parallel postulate (Proposition I.5) to establish that the lines do not converge.^[35] Thus, this alternate interior angle is also 90 degrees, proving that t is perpendicular to l_2. This logical sequence confirms the theorem without contradiction under Euclidean assumptions.^[34] This interaction between perpendicularity and parallel lines finds application in establishing rectangular coordinate systems, where lines parallel to one axis are intersected by perpendicular transversals parallel to the other axis, forming right angles that define the grid.^[36]

Perpendicular Distances

In plane geometry, the perpendicular distance from a point to a line represents the shortest distance between them, achieved along the line segment that forms a right angle with the given line. This concept is fundamental in analytic geometry for measuring separations and projections. For a line expressed in the general form ax + by + c = 0 and a point (x_0, y_0), the perpendicular distance d is calculated using the formula

d = \frac{|a x_0 + b y_0 + c|}{\sqrt{a^2 + b^2}}.

/04:_Similar_Triangles/4.06:_Distance_from_a_Point_to_a_Line) This expression arises from the normalization of the line equation, where \sqrt{a^2 + b^2} is the magnitude of the normal vector (a, b) to the line, and the numerator gives the signed offset from the point to the line along this normal direction.^[37] The derivation relies on identifying the foot of the perpendicular—the point where the normal from (x_0, y_0) intersects the line—as the projection of the point onto the line. The coordinates of this foot can be found by parameterizing the perpendicular line through (x_0, y_0) with direction (a, b) and solving for its intersection with ax + by + c = 0. The distance is then the Euclidean length between (x_0, y_0) and this foot, which simplifies algebraically to the formula above via the dot product or area interpretation (twice the area of the triangle formed by the point and two points on the line divided by the base length).^[37] For instance, consider the line $3x + 4y - 5 = 0 and point (1, 1); substituting yields d = |3(1) + 4(1) - 5| / \sqrt{3^2 + 4^2} = 2/5, confirming the perpendicular separation./04:_Similar_Triangles/4.06:_Distance_from_a_Point_to_a_Line) When considering two parallel lines in the plane, given by ax + by + c_1 = 0 and ax + by + c_2 = 0, the perpendicular distance between them is constant and equals

d = \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}.

^[38] This follows directly from the point-to-line formula: select any point on the first line, where ax + by = -c_1, so its offset to the second line is |-c_1 + c_2| / \sqrt{a^2 + b^2}, independent of the chosen point due to parallelism. The foot of the perpendicular from such a point lies on the second line, establishing the uniform separation. For example, the lines $2x - y + 1 = 0 and $2x - y - 3 = 0 yield d = |-3 - 1| / \sqrt{2^2 + (-1)^2} = 4 / \sqrt{5}.^[37] In the plane, non-parallel lines always intersect at exactly one point, resulting in a shortest distance of zero between them; no common perpendicular exists beyond the trivial case at the intersection.

Perpendiculars in Analytic Geometry

In the Cartesian Plane

In the Cartesian plane, lines are represented in the general form ax + by + c = 0, where (a, b) is the normal vector to the line. Two such lines, ax + by + c = 0 and a'x + b'y + c' = 0, are perpendicular if the dot product of their normal vectors is zero, i.e., a a' + b b' = 0.^[39] This condition arises because the direction vectors of the lines are perpendicular to their respective normals, and orthogonality of the lines corresponds to the normals being parallel to each other's directions. The standard Cartesian coordinate system is an orthogonal coordinate system, where the x-axis and y-axis intersect at right angles, forming the basis for perpendicular measurements in the plane./19%3A_Mathematical_Methods_for_Classical_Mechanics/19.04%3A_Appendix_-_Orthogonal_Coordinate_Systems) In this system, the unit vectors \hat{i} and \hat{j} satisfy \hat{i} \cdot \hat{j} = 0, ensuring that coordinates align with perpendicular directions. To illustrate, consider the line y = 2x + 1, which has slope 2. A line perpendicular to it has slope -1/2, consistent with the product of slopes being -1 for non-vertical lines. The equation of the perpendicular line passing through the point (3, 4) is found using the point-slope form:

y - 4 = -\frac{1}{2}(x - 3)

Simplifying yields y = -\frac{1}{2}x + 5.5, or in standard form, x + 2y - 11 = 0.^[40] Rotations in the Cartesian plane preserve perpendicularity through orthogonal transformation matrices. A 90-degree counterclockwise rotation is given by the matrix

\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix},

which is orthogonal since its transpose equals its inverse, ensuring that dot products—and thus angles of 90 degrees—are maintained.^[41] For vectors \mathbf{u} and \mathbf{v} with \mathbf{u} \cdot \mathbf{v} = 0, the rotated vectors R\mathbf{u} and R\mathbf{v} satisfy (R\mathbf{u}) \cdot (R\mathbf{v}) = 0.^[42]

Foot of the Perpendicular

The foot of the perpendicular from a point P to a line \ell is the intersection point F on \ell where the line through P perpendicular to \ell meets \ell.^[43] This point F serves as the orthogonal projection of P onto \ell and is the unique point on \ell that minimizes the Euclidean distance to P.^[44] Geometrically, for any other point S on \ell, the right triangle PFS (with right angle at F) satisfies the Pythagorean theorem: PS^2 = PF^2 + FS^2, confirming PF as the shortest distance.^[44] In the Cartesian plane, consider a line given by ax + by + c = 0 and a point P(x_0, y_0). The foot F(x, y) lies on the line and on the perpendicular through P, which has direction vector (a, b) (the normal to the line). The parametric equations of this perpendicular are:

x = x_0 + a t, \quad y = y_0 + b t.

Substituting into the line equation yields:

a(x_0 + a t) + b(y_0 + b t) + c = 0,

t = -\frac{a x_0 + b y_0 + c}{a^2 + b^2}.

Thus,

x = x_0 + a t, \quad y = y_0 + b t.

An equivalent closed-form expression is:

x = \frac{b(b x_0 - a y_0) - a c}{a^2 + b^2}, \quad y = \frac{a(-b x_0 + a y_0) - b c}{a^2 + b^2}.

^[37] If the line is parameterized in vector form as \mathbf{Q} + s \mathbf{D} (where \mathbf{Q} = (q_x, q_y) is a point on the line and \mathbf{D} = (d_x, d_y) is its direction vector), the foot from P(x_0, y_0) is \mathbf{Q} + \proj_{\mathbf{D}} (\mathbf{P} - \mathbf{Q}), where the vector projection is:

\proj_{\mathbf{D}} (\mathbf{P} - \mathbf{Q}) = \frac{(\mathbf{P} - \mathbf{Q}) \cdot \mathbf{D}}{\|\mathbf{D}\|^2} \mathbf{D}.

^[45] For example, take the line $3x + 4y - 5 = 0 (so a=3, b=4, c=-5) and point P(1, 1). Compute the denominator a^2 + b^2 = 25 and numerator $3(1) + 4(1) - 5 = 2, yielding t = -2/25. Then:

x = 1 + 3(-2/25) = 19/25, \quad y = 1 + 4(-2/25) = 17/25.

Verification: $3(19/25) + 4(17/25) - 5 = 0, confirming F(19/25, 17/25) lies on the line. The distance PF is |2| / 5 = 2/5, the minimum to the line.^[37]

Applications in Conic Sections

Circles

In circles, a fundamental property of perpendicularity arises from the relationship between the radius and the tangent line at a point of contact. The theorem states that the radius drawn to the point of tangency is perpendicular to the tangent line.^[46] This perpendicularity ensures that the tangent touches the circle at exactly one point, with the radius serving as the normal to the curve at that location.^[47] Another key theorem involving perpendicularity in circles is Thales' theorem, which asserts that if a triangle is inscribed in a circle where one side is a diameter of the circle, then the angle opposite the diameter is a right angle (90 degrees).^[48] This property highlights how the circle's symmetry creates perpendicular angles when a diameter subtends an arc of 180 degrees, forming a semicircle.^[48] For chords intersecting inside a circle, the intersecting chords theorem states that the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.^[46] When the chords are perpendicular, this theorem simplifies calculations in coordinate geometry or optimization problems, such as finding areas or distances, while maintaining the equality of segment products.^[46] In analytic geometry, consider the circle centered at the origin with equation x^2 + y^2 = r^2. The equation of the tangent line at a point (x_1, y_1) on the circle is x x_1 + y y_1 = r^2.^[49] This tangent is perpendicular to the radius vector from the origin to (x_1, y_1), as the normal vector to the tangent line is (x_1, y_1), matching the direction of the radius.^[47]

Ellipses and Parabolas

In conic sections, the normal to an ellipse at any point is defined as the line perpendicular to the tangent at that point, providing a key geometric relation for analyzing curve properties.^[50] The ellipse's reflection property further highlights perpendicularity: a ray from one focus to a point on the ellipse reflects off the tangent such that the reflected ray passes through the other focus, with the normal bisecting the angle between the lines to the foci.^[50] For confocal conics—ellipses or hyperbolas sharing the same foci—the point of intersection of perpendicular tangents, one to each conic, lies on a circle, illustrating a locus related to their shared focal structure.^[51] The standard parametric equations for an ellipse centered at the origin with semi-major axis a along the x-axis and semi-minor axis b along the y-axis are x = a \cos \theta and y = b \sin \theta, where \theta is the eccentric angle.^[52] Differentiating with respect to \theta gives the slope of the tangent as \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} = \frac{b \cos \theta}{-a \sin \theta} = -\frac{b}{a} \cot \theta.^[53] Consequently, the slope of the normal, being the negative reciprocal, is \frac{a}{b} \tan \theta.^[50] For a parabola, the directrix is the line perpendicular to the axis of symmetry, serving as a defining boundary in its geometric construction.^[54] Consider the standard parabola y^2 = 4ax with parametric points (at^2, 2at); the slope of the tangent at parameter t is $1/t. At the endpoints of the latus rectum, corresponding to t = [1](/page/1) and t = -[1](/page/1), the tangent slopes are $1 and -1, respectively, which are perpendicular since their product is -1. More generally, for two points on the parabola with parameters t_1 and t_2, the tangents are perpendicular if the product of their slopes satisfies \frac{1}{t_1} \cdot \frac{1}{t_2} = -1, or equivalently, t_1 t_2 = -1. This condition identifies pairs of points where the tangents intersect at right angles, a useful property in applications like optics and trajectory analysis.^[54]

Hyperbolas

In hyperbolas, perpendicularity manifests prominently in the configuration of asymptotes and tangents. The asymptotes of a hyperbola \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 are the lines y = \pm \frac{b}{a}x, which are perpendicular when their slopes' product is -1, occurring precisely when a = b. This special case defines the rectangular hyperbola, where the asymptotes intersect at right angles at the center.^[55] The rectangular hyperbola can be expressed in rotated coordinates as xy = c^2, with the coordinate axes serving as the perpendicular asymptotes. In this orientation, the transverse and conjugate axes are at 45 degrees to the coordinate axes, each of length 2a (with c^2 = a^2 / 2), emphasizing the symmetry in perpendicular directions from the center. This form highlights the property that the hyperbola equidistantly balances projections along mutually perpendicular axes, with the semi-axes parameters effectively equal in magnitude for the rotated system.^[55] A key application of perpendicularity involves pairs of tangents drawn from an external point (x_1, y_1) to the hyperbola \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. The combined equation of these tangents is given by SS_1 = T^2, where S = \frac{x^2}{a^2} - \frac{y^2}{b^2} - 1, S_1 = \frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} - 1, and T = \frac{xx_1}{a^2} - \frac{yy_1}{b^2} - 1. For the tangents to be perpendicular, the sum of the coefficients of x^2 and y^2 in this equation must vanish, yielding the locus x^2 + y^2 = a^2 - b^2, known as the director circle.^[56] In the rectangular hyperbola where a = b, the director circle degenerates to the origin (x^2 + y^2 = 0), and the perpendicular nature of the asymptotes aligns with the orthogonal axes in the xy = c^2 form, reinforcing the geometric symmetry of the rectangular hyperbola in representing equal scaling in perpendicular directions from the center.^[55]

Perpendiculars in Polygons

Triangles

In a triangle, an altitude is defined as the perpendicular line segment drawn from a vertex to the line containing the opposite side, or to its extension if necessary.^[57] The three altitudes of any triangle are concurrent, meaning they intersect at a single point known as the orthocenter.^[57] This concurrency holds regardless of the triangle's type, highlighting the fundamental role of perpendiculars in triangular geometry. The position of the orthocenter varies with the triangle's angles: in an acute triangle, it lies inside the triangle; in a right triangle, it coincides with the vertex of the right angle; and in an obtuse triangle, it lies outside the triangle. In a right triangle specifically, the two legs form perpendicular sides by definition, and the hypotenuse serves as the diameter of the circumcircle passing through all three vertices.^[58] This property underscores the perpendicular relationship central to right triangles. A key property of altitudes is their use in calculating the triangle's area, given by the formula

\Delta = \frac{1}{2} \times b \times h,

where b is the base length and h is the corresponding altitude (the perpendicular distance from the opposite vertex to the line of the base).^[59] Since each altitude h_a = \frac{2\Delta}{a}, h_b = \frac{2\Delta}{b}, and h_c = \frac{2\Delta}{c} (with a, b, c as the side lengths), the sum of the squares of the altitudes relates directly to the sides via

h_a^2 + h_b^2 + h_c^2 = 4\Delta^2 \left( \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} \right).

This relation connects the perpendicular distances to the side lengths through the area.

Quadrilaterals

In a rhombus, the diagonals are perpendicular to each other and bisect one another at right angles, serving as perpendicular bisectors that divide the rhombus into four congruent right-angled triangles.^[60] This property holds because all sides of the rhombus are equal, and the diagonals are axes of symmetry. In a square, which is a special type of rhombus, adjacent sides are perpendicular, forming right angles at each vertex, while the diagonals are also perpendicular to each other and each diagonal forms a 45-degree angle with the adjacent sides.^[61] These perpendicular relationships contribute to the square's high degree of symmetry, with both diagonals serving as lines of reflection.^[62] A kite is a quadrilateral with two pairs of adjacent congruent sides, and its diagonals are perpendicular, with one diagonal acting as the axis of symmetry that bisects the other diagonal and the two angles adjacent to it.^[63] This perpendicularity of the diagonals divides the kite into two congruent right triangles along the symmetry axis, facilitating its characteristic shape.^[64] For any quadrilateral whose diagonals are perpendicular, the area can be calculated using the formula A = \frac{1}{2} d_1 d_2, where d_1 and d_2 are the lengths of the diagonals; this formula applies directly to shapes like rhombi and kites due to their perpendicular diagonals.^[65] Varignon's theorem states that connecting the midpoints of the sides of any quadrilateral forms a parallelogram, and this parallelogram has perpendicular diagonals (making it a rhombus) if the diagonals of the original quadrilateral are of equal length. This property arises because the sides of the Varignon parallelogram are parallel to and half the length of the diagonals of the original quadrilateral, leading to perpendicularity when those diagonals are congruent.^[66]

Perpendiculars in Higher Dimensions and Vectors

Lines in Three Dimensions

In three-dimensional space, two lines are perpendicular if the dot product of their direction vectors is zero.^[67] This condition extends the two-dimensional concept of perpendicularity, where lines with slopes that are negative reciprocals are orthogonal.^[68] For lines parameterized as \mathbf{r}_1 + t \mathbf{d}_1 and \mathbf{r}_2 + s \mathbf{d}_2, the directions \mathbf{d}_1 and \mathbf{d}_2 satisfy perpendicularity precisely when \mathbf{d}_1 \cdot \mathbf{d}_2 = 0.^[67] Skew lines in three dimensions are non-intersecting lines that are not parallel, lying in distinct planes.^[69] The shortest distance between such lines occurs along their unique common perpendicular, which is the line segment connecting the two skew lines at right angles to both.^[70] This perpendicular direction is given by the cross product of the direction vectors, \mathbf{d}_1 \times \mathbf{d}_2, assuming the lines are not parallel.^[71] The formula for the distance d between skew lines \mathbf{r}_1 + t \mathbf{d}_1 and \mathbf{r}_2 + s \mathbf{d}_2 is

d = \frac{|(\mathbf{r}_2 - \mathbf{r}_1) \cdot (\mathbf{d}_1 \times \mathbf{d}_2)|}{|\mathbf{d}_1 \times \mathbf{d}_2|}

provided \mathbf{d}_1 \times \mathbf{d}_2 \neq \mathbf{0}.^[69] This expression derives from projecting the vector between points on each line onto the common normal direction.^[71] A line is perpendicular to a plane if its direction vector is parallel to the plane's normal vector.^[72] For a plane with normal \mathbf{n} and a line with direction \mathbf{d}, the condition holds when \mathbf{d} = k \mathbf{n} for some scalar k \neq 0, ensuring the line intersects the plane at a right angle.^[73] This property is fundamental in determining orthogonality between lines and surfaces in vector geometry.^[74]

Vector Interpretations

In vector spaces, two vectors \mathbf{u} and \mathbf{v} are orthogonal, or perpendicular, if their dot product satisfies \mathbf{u} \cdot \mathbf{v} = 0.^[75] This condition implies that the vectors form a right angle in the Euclidean plane, extending the geometric notion of perpendicularity to abstract vector representations.^[75] In three dimensions, for orthogonal vectors, the magnitude of their cross product equals the product of their individual magnitudes: |\mathbf{u} \times \mathbf{v}| = |\mathbf{u}| |\mathbf{v}|, since \sin 90^\circ = 1.^[76] An orthonormal basis consists of a set of mutually orthogonal vectors, each of unit length, that spans the vector space.^[77] Specifically, for basis vectors \{\mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_n\}, the conditions are \mathbf{e}_i \cdot \mathbf{e}_j = 0 for i \neq j and \mathbf{e}_i \cdot \mathbf{e}_i = 1 for all i.^[77] Such bases simplify computations in linear algebra, as coordinates with respect to an orthonormal basis preserve lengths and angles via the inner product.^[78] Orthogonality underpins key applications, including orthogonal projections and least squares approximations. The orthogonal projection of a vector \mathbf{b} onto a subspace spanned by orthogonal vectors minimizes the distance \|\mathbf{b} - \mathbf{p}\|, where \mathbf{p} lies in the subspace, leading to solutions in regression and data fitting. In least squares problems, such as A\mathbf{x} \approx \mathbf{b}, the solution \mathbf{x} satisfies the normal equations A^T A \mathbf{x} = A^T \mathbf{b}, where orthogonality of the residual \mathbf{b} - A\mathbf{x} to the column space of A ensures minimality. The Gram-Schmidt process constructs an orthonormal basis from any linearly independent set of vectors, orthogonalizing them sequentially by subtracting projections onto previous vectors and normalizing.^[79] For a set \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}, the first orthonormal vector is \mathbf{e}_1 = \mathbf{v}_1 / \|\mathbf{v}_1\|, and subsequent ones are \mathbf{e}_k = (\mathbf{v}_k - \sum_{i=1}^{k-1} (\mathbf{v}_k \cdot \mathbf{e}_i) \mathbf{e}_i) / \|\cdot\|.^[80] This algorithm is foundational for QR decompositions and numerical stability in solving linear systems.^[80] Perpendicularity generalizes to n-dimensional inner product spaces, where two vectors are orthogonal if their inner product \langle \mathbf{u}, \mathbf{v} \rangle = 0. This framework applies beyond Euclidean spaces to Hilbert spaces, enabling orthogonality in function spaces and infinite dimensions, with orthonormal bases forming complete sets for expansions like Fourier series.

Perpendicularity in Calculus and Functions

Graphs of Functions

In the graphs of functions, the tangent line at a point on a differentiable curve y = f(x) approximates the function locally and has slope given by the derivative f'(x). The normal line at the same point is perpendicular to this tangent and thus has slope equal to the negative reciprocal, -1 / f'(x), assuming f'(x) ≠ 0.^[81]^[82] This relationship follows from the geometric property that two lines with slopes m_1 and m_2 are perpendicular if m_1 \cdot m_2 = -1.^[81] For instance, consider the function f(x) = x^2. At x = 1, the point is (1, 1) and f'(x) = 2x yields a tangent slope of 2. The corresponding normal slope is therefore -1/2. The tangent line equation is y - 1 = 2(x - 1), or y = 2x - 1, while the normal line is y - 1 = (-1/2)(x - 1), or y = (-1/2)x + 3/2. Vertical tangents occur where the derivative f'(x) is undefined or approaches infinity, resulting in a slope that cannot be expressed as a finite number; such tangents are perpendicular to horizontal lines, which have slope 0.^[83] Examples include cusps like y = x^{1/3} at x = 0, where the tangent is vertical despite the function being continuous. In piecewise-defined functions, differentiability at junction points requires the left-hand derivative f'-(c) and right-hand derivative f'+(c) to exist and be equal. However, even if they differ, creating a corner, the approximating tangent segments from each piece can be perpendicular if the product of the one-sided slopes satisfies f'-(c) \cdot f'+(c) = -1.^[84]^[81] This condition ensures the local linear approximations meet at right angles, as in a piecewise linear function where adjacent segments form a 90-degree angle at the breakpoint.

Orthogonal Curves

In geometry, two curves are said to be orthogonal if they intersect at right angles, meaning their tangent lines at the point of intersection are perpendicular. This condition holds when the product of the slopes of the tangents, m_1 and m_2, satisfies m_1 m_2 = -1, provided neither slope is infinite. Orthogonal curves typically arise in families, where one family of curves intersects another family perpendicularly at every intersection point, forming a system of orthogonal trajectories.^[85] To find the orthogonal trajectories of a given family of curves, one first derives the differential equation satisfied by the family and then modifies it to enforce perpendicularity. Suppose the original family satisfies the first-order differential equation \frac{dy}{dx} = f(x, y). The orthogonal family then satisfies \frac{dy}{dx} = -\frac{1}{f(x, y)}, which ensures the tangent slopes are negative reciprocals. Solving this new differential equation yields the equation of the orthogonal trajectories. This method is particularly useful for families defined parametrically or implicitly, as it transforms the problem into a solvable ordinary differential equation.^[85] A classic example involves the family of concentric circles centered at the origin, given by x^2 + y^2 = c^2. Differentiating yields \frac{dy}{dx} = -\frac{x}{y}, so the orthogonal trajectories satisfy \frac{dy}{dx} = \frac{y}{x}, whose solutions are the radial lines y = kx passing through the origin. Each radial line intersects every circle at a right angle, illustrating how lines of constant angle in polar coordinates are orthogonal to circles of constant radius.^[85] Another prominent example is the confocal system of conic sections, where the family of ellipses and the family of hyperbolas sharing the same foci intersect orthogonally. For instance, in the elliptic coordinate system, these confocal conics form an orthogonal grid that tiles the plane without gaps or overlaps.^[86] In complex analysis, orthogonal curves appear naturally as level sets of analytic functions. Specifically, for a holomorphic function f(z) = u(x,y) + iv(x,y), where u and v are the real and imaginary parts, the level curves u(x,y) = c_1 and v(x,y) = c_2 intersect orthogonally because u and v are harmonic conjugates, and their gradients satisfy \nabla u \cdot \nabla v = 0.^[87]^[88] This property underpins the orthogonality of equipotential lines and field lines in two-dimensional electrostatics modeled by analytic potentials.

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