1. Properties of Vector Operations: u, v, w are vectors. a, b are scalars. 0 is the zero vector. 0 is a scalar zero. 1. u + v = v + u 2. (u + v) + w = u + (v + w) 3. u + 0 = u 4. u + (– u) = 0 5. 0 u = 0 6. 1 u = u 7. a (b u ) = (a b ) u 8. a ( u + v ) = a u + a v 9. ( a + b ) u = a u + b u 10. u. v = v. u 11. (au ). v = u. (av ) = a(u. v) 12. u. (v + w) = (u. v) + (u. w) 13. u. u = | u | 2 14. 0. u = 0 15. u & v are if and only if u. v = 0 16. u & v are || if and only if u x v =0 17. (au) x (bv) = (ab)(u x v) 18. u x (v + w) = (u x v) + (u xw) 19. (v + w) x u = v x u + w x u 20. v x u = –(u x v) 21. 0 x u = 0 22. (u x v). w = (v x w). u = (w x u). v 23. (u x v). w = u. (v x w) Vectors Calculus III – Thomas Chapter 12 Sect 1-5 R. M. E. Revised 3-05-07 Lines and Planes in Space Equation for Line thru point P 0 (x 0, y 0, z 0 ) parallel to vector v = v 1 i + v 2 j + v 3 k Vector Form: r(t) = r 0 + t v = r 0 + t |v| (Initial position + time*speed*direction) Parametric Form: x = x 0 + t v 1, y = y 0 + t v 2, z = z 0 + t v 3 Equation for Plane thru point P 0 (x 0, y 0, z 0 ) normal to vector n = Ai + Bj + Ck = Vector Form: n. P 0 P = 0 Component Form: A(x – x 0 ) + B(y – y 0 ) + C(z – z 0 ) = 0 Simplified: Ax + By +Cz = D, where D = Ax 0 + By 0 + Cz 0 Distance from Pt. S to a Line Through P parallel to vector v: d = Distance Pt. to plane: d = PS. between planes:  = cos -1 ( ) Vector (v) Parallel to two intersecting planes with normals n 1 and n c : v = n 1 x n 2 v |v| | PS x v | | v | n |n| n 1. n 2 |n 1 | |n 2 | Vector Applications Work (W) by constant force F throught displacement D: W = F. D = | F | | D | cos  Torque by applying force vector F throught lever arm (vector) r Torque vector = r x F = ( | r | | F | sin  ) n [Magnitude = | r | | F | sin  ] Area of a Parallelogram formed by vectors u, v, -u, -v = | u x v | Area of a Triangle formed by vectors u, v, = ½ | u x v | Volume of a Parallelepiped formed by vectors u,v, and w = ( u x v ). w Vector Operations: Vectors u =, v = and n normal to u & v. Scalar k Addition: u + v = Scalar Multiplication: ku = Dot Product (scalar): u. v = u 1 v 1 + u 2 v 2 + u 3 v 3 Angle between two vectors:  = cos -1 ( ) or  = cos -1 ( u. v / ( | u | | v | ) ) Cross Product(vector): u x v = ( | u | | v | sin  ) n i j k u x v = u 1 u 2 u 3 = i – j + k v 1 v 2 v 3 u 2 u 3 u 1 u 3 u 1 u 2 v 2 v 3 v 1 v 3 v 1 v 2 u 1 v 1 + u 2 v 2 + u 3 v 3 | u | | v | Vector Definitions: Vector: Directed Line Segment. Directed line segment AB has Initial Point A (x A, y A, z A ), Terminal Point B(x B, y B, z B ). Vector AB is written as (x B – x A )i + (y B – y A )j + (z B – z A )k or Length denoted |AB|. Vectors are equal if they have the same length and direction Standard Unit Vectors: i = j = k = Magnitude (or Length) | v | (a scalar) : | v | = v 1 2 + v 2 2 + v 3 2 = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 + (z 2 – z 1 ) 2 Unit vector in the direction of v is If v ≠ 0 then v = | v | or its magnitude times its direction Midpoint of a lne segment connecting points P 1 = (x 1, y 1, z 1 ) and P 2 = (x 2, y 2, z 2 ) is ( ) Orthogonal: Perpendicular Vector Projection of u onto v: proj v u = ( ) v = ( )( ) Scalar component of u in the direction of v: | u | cos  = = u. ( ) Writing u as a Vector Parallel to v plus a Vector Orthogonal to v u = proj v u + ( u – proj v u ) u = ( ) v + ( u – ( ) v ) Parallel to v Orthogonal to v Normal: Unit vector n is normal (perpendicular) to the plane formed by two non-parallel vectors u & v. The direction of n is determined by the right hand rule. Use the fingers or your right hand to push u into v and your thumb will point in the direction of n. Triple Scalar or Box Product (scalar): (u x v). w |(u x v). w| = | u x v | | w | |cos  | x 1 +x 2 y 1 +y 2 z 1 +z 2 2 2 2,, v | v | u. v | v | v | v | u. v | v | 2 v | v | u. v | v | u. v | v | 2 n = = u x v |u x v| u x v |u| |v| sin 