Math Prelim 2 Cheat Sheet - Vector Operations Dot Product A B A B cos A B AxBx AyBy AzBz Direction of A A A Magnitude A Ax 2 Ay 2 Az 2 i Cross Product j

Math Prelim 2 Cheat Sheet - Vector Operations Dot Product A...

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Vector Operations Dot Product: cos A B A B A B AxBx AyBy AzBz θ = = + + Direction of A: A A Magnitude: ( 29 2 2 2 A Ax Ay Az = + + Cross Product: sin i j k Ay Az Ax Az Ax Ay A B Ax Ay Az i j k A B n By Bz Bx Bz Bx By Bx By Bz θ = = - + = Applications: 0 A B B = (A and B are orthogonal) 0 A B B = (A and B are parallel) Area of Triangle ABC = 2 AB AC B uuur uuur Area of Parallelpiped = ( ) A B C B (A & B are on the base, C is upwards) Projection of B onto A = ( 29 cos A A B A A B A proj B B A A A A A A θ ° = = = ° Scalar component of B in direction of A = ( 29 cos B A B A θ = Angle between two vectors: 1 cos A B A B θ - ° = B as sum of orthogonal and parallel vectors to A: ( ) A A B proj B B proj B = + - Lines and Planes in Space: Vector from point A to point B: ( ) ( ) ( ) AB Bx Ax i By Ay j Bz Az k = - + - + - uuur Line from a point and a vector: ( , , ) o o o P x y z v ai bj ck = + + r : o o o x x at L y y bt z z ct = + = + = + Distance from point to a line: : o o o x x at L y y bt z z ct = + = + = + 1 1 1 0 0 0 ( , , ) ( , , ) S x y z P x y z v ai bj ck = + + r PS v

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