Angles Between Vectors Orthogonal Vectors The Dot Product Angles Between Vectors Orthogonal Vectors Section 6.2a HW: p. 520 13-27 odd

Definition: Dot Product The dot product or inner product of u = u , u and v = v , v is 1 2 1 2 u v = u v + u v 1 1 2 2 The sum of two vectors is a… vector! The product of a scalar and a vector is a… vector! The dot product of two vectors is a… scalar!

Properties of the Dot Product Let u, v, and w be vectors and let c be a scalar. 1. u v = v u 2 2. u u = |u| 3. 0 u = 0 4. u (v + w) = u v + u w (u + v) w = u w + v w 5. (cu) v = u (cv) = c(u v)

Finding the Angle Between Two Vectors (Law of cosines) v – u v u

Angle Between Two Vectors Theorem: Angle Between Two Vectors v – u If 0 is the angle between nonzero vectors u and v, then v u and

Definition: Orthogonal Vectors The vectors u and v are orthogonal if and only if u v = 0. The terms “orthogonal” and “perpendicular” are nearly synonymous (with the exception of the zero vector)

Guided Practice Find each dot product = 23 = –10 = 11 1. 3, 4 5, 2 1. 3, 4 5, 2 = –10 2. 1, –2 –4, 3 3. (2i – j) (3i – 5j) = 11

Guided Practice u v = 0!!! Prove that the vectors u = 2, 3 and v = –6, 4 are orthogonal Check the dot product: u v = 0!!! Graphical Support???

First, let’s look at a brain exercise… Page 520, #30: Find the interior angles of the triangle with vertices (–4,1), (1,–6), and (5,–1). Start with a graph… A(–4,1) B(5,–1) C(1,–6)

First, let’s look at a brain exercise…

First, let’s look at a brain exercise…

First, let’s look at a brain exercise…

Definition: Vector Projection The vector projection of u = PQ onto a nonzero vector v = PS is the vector PR determined by dropping a perpendicular from Q to the line PS. Q u P S v R Thus, u can be broken into components PR and RQ: u = PR + RQ

Definition: Vector Projection Q u P S R Notation for PR, the vector projection of u onto v: PR = proj u v The formula: u v proj u = v v |v| 2

Some Practice Problems Find the vector projection of u = 6, 2 onto v = 5, –5 . Then write u as the sum of two orthogonal vectors, one of which is proj u. v Start with a graph…

u = proj u + u = 2, –2 + 4, 4 Some Practice Problems… Find the vector projection of u = 6, 2 onto v = 5, –5 . Then write u as the sum of two orthogonal vectors, one of which is proj u. v Start with a graph… u = proj u + u = 2, –2 + 4, 4 v 2

Some Practice Problems… Find the vector v that satisfies the given conditions: 2 u = –2,5 , u v = –11, |v| = 10 A system to solve!!!

Some Practice Problems… Find the vector v that satisfies the given conditions: 2 u = –2,5 , u v = –11, |v| = 10 OR