Calculus III Chapter 12 Br. Joel Baumeyer Christian Brothers University

Displacement Vectors A displacement vector from one point to another is an arrow with its tail at the first point and its tip at the second. The magnitude or length of the displace- ment is the distance between the points, and is represented by the length of the arrow. The direction of the displacement vector is the direction of the arrow. Displacement vectors which point in the same direction and have the same magnitude are considered to be the same, even if they do not coincide.

Vectors and Ordered n-tuples The standard basis displacement vectors: I, j and k are represented by the ordered triples [1,0,0], [0,1,0] and [0,0,1] respectively. 0 = [0,0,0]. If v =, w =, u = then the following are provable: (a,b - scalars) u + v = v + u (a +b)v = av + bv (u + v ) + w = u + (v + w )a(v + w) = av + aw a(bv) = (ab)v v + 0 = v 1v = v v + w = + = av = a = Norm of a vector: ||v|| =

The Dot Product & Properties Definition: v  w = ||v|| ||w|| cos ,  the angle between v and w. Thus  = If then, v  ( w) = (v  w) = ( v)  w; v  w = w  v (v + w)  u = v  u + w  u; v  v = ||v|| ||v|| A unit vector u : A vector whose norm (magnitude) is 1. u = v/||v||

Applications of the Dot Product Work is an example of the dot product: W = F d Given two vectors F and d, with d not 0, then is parallel to d and is perpendicular to d. Then : and The equation of the plane with normal vector n = ai +bj +ck and containing the point is Or ax + by + cz = d, if

Cross Product & Properties Definition: v  w = ||v|| ||w|| sin  n, n a unit vector perpendicular to the plane formed by v and w. i  j = k,, j  k = i, k  i = j, j  i = -k, k  j = -i, i  k = -j i  i = 0, j  j = 0, k  k = 0 v  ( w) = (v  w) = ( v)  w v  w = -w  v; u  (v + w) = u  v+ u  w.

Applications of the Cross Product The cross product v  w is a vector that has magnitude equal to the area of the parallelo- gram determined by v and w. direction perpendicular to plane of v and w and determined by the right-hand rule. The volume of a parallelepiped (box) formed by the three vectors u, v and w is given by |u  (v  w)| = = |v  (u  w)| = |w  (v  u)|