What is Perpendicular lines: Definition and 19 Discussions

In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees). The property extends to other related geometric objects.
A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order.
Perpendicularity easily extends to segments and rays. For example, a line segment



{\displaystyle {\overline {AB}}}
is perpendicular to a line segment



{\displaystyle {\overline {CD}}}
if, when each is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols,





{\displaystyle {\overline {AB}}\perp {\overline {CD}}}
means line segment AB is perpendicular to line segment CD. For information regarding the perpendicular symbol see Up tack.
A line is said to be perpendicular to a plane if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines.
Two planes in space are said to be perpendicular if the dihedral angle at which they meet is a right angle (90 degrees).
Perpendicularity is one particular instance of the more general mathematical concept of orthogonality; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its normal.

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  1. L

    I Two vectors and two perpendicular lines

    In ##\mathbb{R}^2##, there are two lines passing through the origin that are perpendicular to each other. The orientation of one of the lines with respect to ##x##-axis is ##\psi \in [0, \pi]##, where ##\psi## is uniformly distributed in ##[0, \pi]##. Also, there are two vectors in...
  2. M

    B How to calculate a perpendicular line?

    Hi, i get the math that is involved but if I have only the x,y coordinates for 2 points to connect and if i want to know what will be the perpendicular line to the line connecting these two points, how can I do that?
  3. H

    Shortest distance between a line and a point?

    Homework Statement find the shortest distance from (0,0) to the line passing A(2,3) and B(3,5) Homework Equations ## \frac{y-y1}{y2-y1} = \frac{x-x1}{x2-x1} ## y-y1 = m (x-x1) m1 * m2 = -1 (m1 perpendicular to m2) The Attempt at a Solution line passing A and B points ## \frac{y-3}{5-3} ## =...
  4. M

    Are Skew Lines Considered Perpendicular in 3D with Parallel Vectors?

    in 3D assume two skew lines L1: x=x°+at, y=y°+bt, z=z°+ct, L2: x=x°°+ds, y=y°°+es, z=z°°+fs. therefore L1, L2 parallel vectors are respectively: v1= <a, b, c>, v2= <d, e, f>. if v1.v2= ad+be+cf= 0 (vectors are perpendicular), are the line L1, L2 considered perpendicular also or beside the dot...
  5. T

    Product of the gradients of perpendicular lines proof help

    Okay I'm having a little trouble understanding a section of this proof about the product of the gradients of perpendicular lines given in my textbook. I'm going to type the proof out but there will be a link at the bottom to an online version of the textbook so you can see the accompanying...
  6. EnchantedEggs

    Why Are These Lines Perpendicular?

    Hi all, When you have a surface defined by F(x, y, z) = 0 where x = f(t), y= g(t) and z= h(t) and a point on this surface P_0 = (x_0, y_0, z_0) , could someone explain to me why a line through P_0 with direction numbers [\frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt}] is perpendicular to a...
  7. MarkFL

    MHB Perpendicular lines and the product of their slopes

    As students taking pre-calculus, you are probably aware that when two lines are perpendicular, the product of their slopes is -1. Let's see why this is. Let one line be $\displaystyle y_1=m_1x+b_1$ and the other line be $\displaystyle y_2=m_2x+b_2$. Now,we know the angle of inclination of a...
  8. H

    Proving Perpendicular Lines in a Quadrilateral

    Homework Statement ABCD is a quadrilateral with AB=CD P,Q,R,S are the midpoints of AD,AC,BD,BC respectively Prove that PS is perpendicular to QR Homework Equations The Attempt at a Solution I've tried a bunch of different avenues, but I think that the most promising one is...
  9. N

    Relation Between Complex Slopes of Perpendicular Lines

    Homework Statement what is the relation between complex slopes of two perpendicular lines?? :think: Homework Equations The Attempt at a Solution i think it is same as the slopes of two st. lines that is the product shud be -1...but the answer given is sum of slopes is 0...
  10. S

    Mobius Maps: Parallel and Perpendicular Lines, Disjoint Circles

    Homework Statement Suppose T is a Mobius map take Real -> Real and infinite infinitely to 0 a) What's the image of the family of lines parallel to Real? b) What's the image of the family of lines perpendicular to Real? c) Show the Mobius map take D = {z :|z|<1} onto itself iff Tz =...
  11. T

    Perpendicular Lines: Slope, Angle & Equations

    A very simple and noob question, sorry for this. Looking at http://en.wikipedia.org/wiki/Perpendicular I found that two lines are perpendicular if and only if the product of their slopes is -1. If I have two lines described by the following equations: y = ax + b g = cx + d then...
  12. F

    Finding Perpendicular Lines to y=0.25x and Tangent to f(x)=1/x

    Homework Statement Find two straight lines that are perpendicular to y=0.25x and tangent to the curve f(x) = 1/x. Homework Equations y=0.25x f(x) = 1/x. The Attempt at a Solution What I did was equate y and f(x) and determined when they equal which is 2 and -2. The points are (2, 1/2) and...
  13. B

    Solve Perpendicular Lines: A(1,2), B(4,0), C(k,2), D(1,-3)

    Determine a constant real number k such that the lines AB and CD are perpendicular. A(1,2), B(4,0), C(k,2), D(1,-3). (answer given is k=-3/2) If two lines are perpendicular the product of their slopes is -1. The slope of AB is \frac{0-2}{4-1} = \frac{-2}{3} The slope of CD is...
  14. D

    Really easy question about perpendicular lines in 3D?

    Homework Statement Find the line through (3, 1, -2) that intersects and is perpendicular to the line x = -1 + t, y = -2 + t, z = -1 + t. Homework Equations line l through P (x1, y1, z1) and Q (x2, y2, z2) has the following form x = x1 + (x2 - x1)t y = y1 + (y2 - y1)t z = z1 + (z2 -...
  15. P

    Find slope of 2 perpendicular lines

    Homework Statement Find the values of k such that lines 3kx+8y = 5 and 6y -4kx = -1 are perpendicular. I don't need the answer, but a push in a right direction. However, feel free to solve the equation :) Homework Equations 1. Slope of a line is m = [y2-y1]/[x2-x1] where we have 2...
  16. Y

    Perpendicular Lines in 3 Space

    i am given parametric equations of 2 lines L1 and L2 and i am asked to find the vector, parametric and symmetric equation of a line that intersects these 2 lines at 90 degrees so a line that is perpendicular to those 2 lines. how do i go about this? at first i tried converting the parametric...
  17. A

    Slope of Perpendicular Line to x - 3y = 9

    I don't know if I'm having a brain fart or what but I'm drawing a blank...:smile: What is the slope of a line perpendicular to x - 3y = 9? Notes: Perpendicular lines have slopes that are negative reciprocals of each other. Example: So, if the original line has a slope of -3/2, the...
  18. D

    Perpendicular Lines in Three Space

    Hello, Please bear with me...my brain is in vapor lock. I have a line L1 given by the following parametric equations: x = 2+3t, y = -1+5t, z = 8+2t I need to find the equation of a line L2 passing through point B = (1,2,5) and perpendicular to L1. For the life of my tired, worn...
  19. cokefree

    Find Perpendicular Lines Algebraically: Tips & Tricks

    Could some tell me what's the other way you could find if lines are perpendicular, other than graphing. I know another way to do it, and its algebracaly, but there's another way you could do it algebracaly too.