8_Lines_Planes - Lines and Planes Lines Let l be a line parallel to vector v v = v1 i v 2 j v 3 k Let P0(a b c be a point on the line and P x y z be any

# 8_Lines_Planes - Lines and Planes Lines Let l be a line...

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Lines and Planes Lines Let l be a line parallel to vector v . v v 1 i v 2 j v 3 k Let P 0 ( a , b , c ) be a point on the line and P ( x , y , z ) be any other point on the line. P 0 P is a vector parallel to the line. P 0 P v so The parametric equations of a line are: x a v 1 t y b v 2 t z c v 3 t
To find the equation of a line you need a point on the line and avector parallel to the line.ex. Find the equation of the line through P(1, 5, 3) parallel tovector v2, 0, 12ikex. Find the equation of the line through points P(1, 2, 3) andQ(3, 6, 9). .
ex. Find a vector parallel to the line x 9 - t y 3 z 6 t ex. Determine which of the following line equations represent the same line as x 9 - t y 3 z 6 t x 1 7 2 t x 2 11 - 3 t x 3 9 t y 1 3 y 2 3 y 3 3 z 1 12 - 12 t z 2 10 18 t z 3 6 t
Planes Let P 0 ( a , b , c ) be a point in the plane. If P(x, y, z) is any other point in the plane, then P 0 P is a vector that lies in the plane. Let n n 1 i n 2 j n 3 k be a vector perpendicular (normal) to the plane. Then
vector normal (perpendicular) to the plane.ex. Find the equation of the plane through P(2, 6, 3) which isnormal to n3, -2, 1ex. Find the equation of the plane through P(2, 6, 3), Q(-1, 0, 1)and R(4, 4, 4). n .
ex. Find a normal vector to the plane z2xy1ex. Determine if the following two lines are parallel,perpendicular, intersect at a non right angle, or are skew. . x 1 3 5 t x 2 2 s y 1 1 - 5 t y 2 - 7 - s z 1 10 t z 2 3 - 2 s
ex. x 1 t x 2 1 s y 1 2 2 t y 2 4 4 s z 1 3 - t z 2 2 2 s
Do: 1. Find the equation of the line through P(1,1,1) perpendicularto the plane 2x-3y5z42. Find the equation of the plane through P(2, 5, 1) parallel to theplane z = -3.3. Find the equation of the line of intersection of planesx + y + z = 1 and x – 2z = 0.4. Is the line x-12t,y-2t,zthe same line asx212t,y2t,z? . 2
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