Ch 12.5: Equations of Lines and PlanesGoal: Find a vector and parametric equation of a line L which goesthrough the point P0 = (x0, y0, z0) and is parallel to the vectorv =< a, b, c >.Notation: Let r0 =< x0, y0, z0 > and r =< x , y , z > where(x , y , z) is a general point on L.

SummaryI A vector equation of a line L which goes through the point

P0 = (x0, y0, z0) and is parallel to the vector v =< a, b, c > is

I Parametric equations of a line L which goes through the pointP0 = (x0, y0, z0) and is parallel to the vector v =< a, b, c > is

ExampleFind a vector equation and parametric equations for the line thatpasses through the point (1, 2, 3) and is parallel to the vector< 2, 3,−1 >.

Symmetric equationsLet the vector equation of a line L is given by

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

That is, the line L goes through the point and isparallel to the vector .The corresponding parametric equation is

Solving for t, one get what are called the symmetric equations:

ExampleFind parametric equations and symmetric equations of the linethat passes through the points A = (2, 3,−1) and B = (3, 1, 1). Atwhat point does this line intersect xy -plane? xz-plane?

Line segmentBy choosing a finite interval for t, one can obtain a line segment.The line segment from the tip of the vector r0 to the tip of thevector r1 is given by the equation

r(t) = (1− t)r0 + t r1 0 ≤ t ≤ 1

PlanesGoal: To find a vector equation of the plane which goes through apoint P0 = (x0, y0, z0) which is normal (i.e. orthogonal) to a vectorn =< a, b, c >.

Formula - vector equation of a planeLet P0 = (x0, y0, z0) be a fixed point on the plane andP = (x , y , z) be a general point on the plane. If r0 and r be theposition vectors of P0 and P, respectively, and if n a vector normalto the plane, then n is normal to r− r0. That is

n · (r− r0) = 0

⇒ n · r = n · r0

Scalar and linear equation of the planeLetting r0 =< x0, y0, z0 >, r =< x , y , z >, and n =< a, b, c >, wecan rewrite the following vector equation as a scalar equation.

n · (r− r0) = 0

⇔ < a, b, c > · (< x , y , z > − < x0, y0, z0 >) = 0

⇔ < a, b, c > · (< x − x0, y − y0, z − z0 >) = 0

⇔⇔

Example 4Find an equation of the plane through the point (2, 4,−1) withnormal vector n =< 2, 3, 4 >. Find the intercepts and sketch theplane.

Example 5Find an equation of the plane through the point P = (1, 3, 2),Q = (3,−1, 2), and R = (1, 2, 0).

ExampleFind the point at which the line with parametric equationsx = 2 + t, y = −2t, z = 1 + t intersects the plane x + 5y −2z = 7.

Examples1. Find the angles between the planes x + y + z = 1 and

x − 2y + 3z = 1.2. Find symmetric equations for the line of intersection of these

two planes.

Distance∗ from a point to a planeGoal: Verify that the shortest distance D from a pointP1 = (x1, y1, z1) to the plane ax + by + cz + d = 0 is given by

D =ax1 + by1 + cz1 + d√

a2 + b2 + c2.

Some notation: Let P0 = (x0, y0, z0) be a fixed point in the plane.

ExampleFind the distance between the parallel planes x + 2y − 2z = 1 and4x − y + z = 2.

Class ExerciseFind symmetric equations for the line of intersection of these twoplanes.

4x − 3y − 3z = −5, 3x + y + z = 6