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Chapter 12 – Vectors and the Geometry of Space12.5 Equations of Lines and Planes

12.5 Equations of Lines and Planes

Objectives: Find vector, parametric,

and general forms of equations of lines and planes.

Find distances and angles between lines and planes

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Lines in 2DA line in the xy-plane is

determined when a point on the line and the direction of the line (its slope or angle of inclination) are given.

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Lines in 3DA line L in 3D space is

determined when we know:◦A point P0(x0, y0, z0) on L

◦The direction of L, given by a vector

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Definition – Vector EquationSo, we let v be a vector parallel to L. Let P(x, y, z) be an arbitrary point on L. Let r0 and r be the position vectors of P0 and P.

If a is the vector with representation from P to Po.

Then the Triangle Law for vector addition gives r = r0 + a.

However, since a and v are parallel vectors, there is a scalar t such that a = tv.

So we have

This is a vector equation of L.

1 r = r0 + t v

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Definition continuedWe can also write this definition in component form:

r = <x, y, z> and r0 = <x0, y0, z0>

◦ So, vector Equation 1 becomes:

Positive values of t correspond to points on L that lie on one side of P0.

Negative values correspond to points that lie on the other side.

<x, y, z> = <x0 + ta, y0 + tb, z0 + tc> 2

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Definition continued Two vectors are equal if and only if corresponding

components are equal.

Hence, we have the following three scalar equations.

x = x0 + at y = y0 + bt z = z0 + ct, where t ℝ These equations are called parametric equations of

the line L through the point P0(x0, y0, z0) and parallel to the vector v = <a, b, c>.

Each value of the parameter t gives a point (x, y, z) on L.

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Example 1 – pg 824 #2Find a vector equation and

parametric equations for the line.

The line through the point (6,-5,2) and parallel to the vector <1,3,-2/3>.

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Definition - Symmetric Equations If we solve the equations for t

x = x0 + at, y = y0 + bt, z = z0 + ct,

we get the following symmetric equations.

0 0 0x x y y z z

a b c

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Example 2Find parametric equations and

symmetric equations for the line.

The line through the points (6,1,-3) and (2,4,5).

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Equations of Line SegmentsThe line segment from r0 to r1 is

given by the vector equation

where 0 ≤ t ≤ 1

r(t) = (1 – t)r0 + t

r1

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Definition – Skew Lines

Lines that are skew are not parallel and do NOT intersect. They do NOT lie in the same plane.

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Example 3Determine whether the lines L1

and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection.

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2

1 3 2:2 2 12 6 2

:1 1 3

x y zL

x y zL

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PlanesA plane in space is determined

by:◦A point P0(x0, y0, z0) in the plane

◦ A vector n that is orthogonal to the plane

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Definition – Normal VectorThis orthogonal vector n is called

a normal vector.The normal vector n is

orthogonal to every vector in the given plane.

In particular, n is orthogonal to r – r0.

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Equation of PlanesIf n is orthogonal to r – r0 we

have the following equations:

Either Equation 5 or Equation 6 is called a vector equation of the plane.

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Equations of PlanesThis equation is the scalar equation of

the plane through P0(x0, y0, z0) with normal vector n = <a, b, c>.

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Linear Equation

where d = –(ax0 + by0 + cz0)

◦This is called a linear equation in x, y, and z.

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Example 4 – pg. 825 # 34Find an equation of the plane.

The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1 + t, z = 2 – t.

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Example 5 – pg. 825 # 45Find the point at which the line

intersects the given plane.

x = 3 – ty = 2 + tz = 5tx – y + 2z = 9

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Example 6 – pg. 825 #48Where does the line through

(1,0,1) and (4, -2, 2) intersect the plane x + y + z = 6?

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More Examples

The video examples below are from section 12.5 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 3◦Example 4 ◦Example 7

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Demonstrations

Feel free to explore these demonstrations below.

Constructing Vector Geometry Solutions