7_Lines_Planes - Lines and Planes Lines Let Let P0(a b c be a point on the line and P(x y z be any other point on the line l be a line parallel to

7_Lines_Planes - Lines and Planes Lines Let Let P0(a b c be...

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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 a vector parallel to the line. ex. Find the equation of the line through P(1, 5, 3) parallel to vector v = 2 , 0 , 1 = 2 i + k . ex. Find the equation of the line through points P(1, 2, 3) and Q(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
The equation of a plane is n 1 x + n 2 y + n 3 z = n p where n is the normal vector and p is the vector formed by the point. To find the equation of a plane you need a point on the plane and a vector normal (perpendicular) to the plane.

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