6.2 Dot Products of Vectors

I. Dot Products A.) Def. - A scalar quantity representing the multiplication of two vectors used to calculate the angle between two vectors.

B.) Properties

C.) Ex. 1 – Find each dot product.

D.) Ex.2- Use the dot product to find the length of v.

II. Angles Between Vectors A.) Thm: If θ is the angle between two non-zero vectors u and v, then Proof is in the book. KNOW IT!!!

B.) Ex. 3 – Find the angle between vectors u and v.

C.) Vectors are perpendicular if D.) Vectors are orthogonal iff (Note: the zero vector is orthogonal to every vector in the plane, but not considered perpendicular.)

II. Vector Projection A.) The projection of u = onto v = is the vector determined by dropping a perpendicular from Q to Q u P R S v

The notation for the projection of u onto v is given by the following: Q S R u The notation for the projection of u onto v is given by the following:

B.) Def- The projection of u onto v if u and v are both non-zero vectors is

C.) Ex. 4 - Find the vector projection of u onto v if .

D.) Now, write u2 as the sum of two orthogonal vectors, one of which is the