8_Lines_Planes_(notes)

8_Lines_Planes_(notes) - vector normal (perpendicular) to...

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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
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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).
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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
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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
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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
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Unformatted text preview: vector normal (perpendicular) to the plane. ex. Find the equation of the plane through P(2, 6, 3) which is normal to n 3, -2, 1 . ex. Find the equation of the plane through P(2, 6, 3), Q(-1, 0, 1) and R(4, 4, 4). ex. Find a normal vector to the plane z 2 x y 1 . ex. 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) perpendicular to the plane 2 x-3 y 5 z 4 . 2. Find the equation of the plane through P(2, 5, 1) parallel to the plane z = -3. 3. Find the equation of the line of intersection of planes x + y + z = 1 and x 2z = 0. 4. Is the line x-2 1 2 t , y-2 t , z the same line as x 2 1 2 t , y 2 t , z ?...
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This note was uploaded on 03/29/2008 for the course MATH 1224 taught by Professor Dontremember during the Spring '08 term at Virginia Tech.

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8_Lines_Planes_(notes) - vector normal (perpendicular) to...

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