12.3: Dot Product and the Angle Between Two Vectors

University:
The Pennsylvania State University
Course:
Math 231 | Calculus Severable Variables
Academic year:

2024
Views:

352

Pages:

2
Author:

Alexis H.

12.3 Dot Product and the Angle between two vectors We can compute the parallel part by projecting w onto a vector parallel to the bridge. Finding projections of one vector onto another is easily done with the dot product In words, to compute the dot product, multiply the corresponding components and add First, the dot product is commutative: v ¢ w = w ¢ v Definition: because the components can be multiplied in either order. Second, Dot Product the dot product of a vector with itself is the square of the length: The dot product v w of two v = hv1; v2; v3i vectors v = hv1; v2; v3i v ¢ v = v1v1 + v2v2 + v3v3 = v12 + v22 + v32 = jjvjj w = hw1; w2; !3i is the scalar defined by 2 Theorem 1: v ¢ ! = v1w1 + v2w2 + v3!3 0¢v =v¢0=0 commutativity: v ¢ w = w ¢ v (¸v) ¢ w = v ¢ (¸w) = ¸ (v ¢ w) pulling out scalars: u ¢ (v + w) = u ¢ v + u ¢ w distributive law: 2 relation with length: v ¢ v = kvk This angle is not uniquely defined because both the angle and 2 pi minus the angle can serve as an angle between v and w. Furthermore, any multiple of 2 pi may be added to the angle. All of these angles have the same cosine, so it does not matter which angle used Theorem 2: Dot product and the angle v ¢ w = kvk kwk cos µ or cos µ = v¢w kvk kwk Two nonzero vectors v and w are called perpendicular or orthogonal if the angle between them is 90 degrees The standard basis vectors are mutually orthogonal and have length 1. In terms of dot products, because i = <1, 0, 0>, j = <0, 1, 0>, and k = <0, 0, 1>, i¢j =j¢k =j¢k =0 i¢i=j¢j =k¢k =1 The angle between v and u is obtuse if and only if the dot product is less than zero. The angle between v and u is acute if and only if the dot product is greater than zero Another important use of the dot product is in finding the projection of a vector u along a nonzero vector v. The projection is the vector parallel to v obtained by dropping a perpendicular from u to the line through Projection of u along v: ujjv = µ u¢v kvk ¶ u = ujjv + uv ev