Prelim #1 Cheat Sheet - : L: , , , ( , , ), ( , , ), , The...

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( 29 1 1 2 2 3 3 2 Dot Product: cos and are orthogonal vectors if 0 cos Scalar component in direction of cos To Vecto e r Operati xp o r ns: v u v u v u v u v u v u v u v v u v proj u u v v v v u v u u v θ = = + + = = = = = r r r r g r r r r g r r r g r r r r r r r r r r g r ( 29 ( 29 1 2 3 1 2 3 ess as a Vector Paralell to plus a Vector orthogonal to : Cross Product: sin Area of Parallelogram = sin sin Volume of v v u v v u proj u u proj u u v u v n u v u v n u v i j k u v u u u v v v = + - = = = = r r r r r r r r r r r r r r r r r r r r r ( 29 -1 0 0 0 0 Parallelepiped = cos area of base height Angle Between Two Vecto Lines and rs: = cos Parameterizing Equation Planes in Space s of a Line through P ( , , ) p u v w u v w u v u v x y z = = r r r r r r g g r r g r r 1 2 3 0 1 0 1 0 3 0 0 0 0 0 arallel to : , y , z Vector from Point A to Point B: ( ) ( ) ( ) Line from a Point and a Vector: P ( , , ) , L: v v i v j v k x x tv y tv z tv AB Bx Ax i By Ay j Bz Az k x y z v Ai Bj Ck x x A = + + = + = + = + = - + - + - = + + = + r r r r r r uuur r r r r r r r { } { } 0 0 0 0 0 0 0 0 0 0 0 0 , , Distance from a Point to a Line Through Parallel to
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Unformatted text preview: : L: , , , ( , , ), ( , , ), , The plane through ( , , ) no t y y Bt z z Ct S P v PS v x x At y y Bt z z Ct S x y z P x y z v Ai Bj Ck d v P x y z = + = + = + = + = + = + + = r uuur r r r r r r rmal to : ( ) ( ) ( ) A vector parallel to the Line of Intersection of Two Planes: Take the cross product of their normals Distance from a Point to a Plane : n Ai Bj Ck A x x B y y C z z Ax By Cz D S d PS = + +-+-+-= + + = = r r r r u r is a point on the plane (intercept) and is normal to the plane n n S n Ai Bj Ck = + + r uu g r r r r r...
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