1. 12.3 The Dot Product

2. The dot product of u and v in the plane is The dot product of u and v in space is Two vectors u and v are orthogonal  if they meet at a right angle.  if and only if u ∙ v = 0 (since slopes are opposite reciprocal) (Read “u dot v”) Definitions Examples

3. Another form of the Dot Product: Properties

4. Find the angle between vectors u and v : Examples

5. Angles between a vector v and 3 unit vectors i, j and k are called direction angles of v, denoted by α, β, and γ respectively. Since we obtain the following 3 direction cosines of v: Direction Cosines

6. Let u and v be nonzero vectors.  w is called the vector projection of u onto v, denoted by proj v u  The signed magnitude of the vector projection is called the scalar projection of u onto v, or vector component of u along v, denoted by comp v u w u v Projections

7. 1) Compute 2) Compute 4) Find the angle between vectors v and w. 3) List all pairs of orthogonal and/or parallel vectors. 6) Find scalar projection of w onto u. 5) Find the unit vector in the direction u. 7) Find vector projection of w onto u. Examples