Dot Products Vector Projections Honors PreCalc 8 3

Dot Products & Vector Projections Honors Pre-Calc 8. 3

Dot Product of Vectors in a Plane • Dot product of a ∙ b = a 1 b 1 + a 2 b 2 and is defined as Unlike vector operations we did yesterday, final answer is not in vector form but is a scalar instead. If two vectors have a dot product equal to zero then the two vectors are said to be orthogonal or perpendicular to each other. ***The only time orthogonal does not mean perpendicular is when dealing with the zero vector. The zero vector is orthogonal to all vectors.

Example 1 Find the dot product of u and v. Then determine if u and v are orthogonal. a) b)

Properties of the Dot Product • Commutative Property: u∙v=v∙u • Distributive Property: u ∙ (v + w) = u ∙ v + u ∙ w • Scalar Multiplication Property: k(u ∙ v) = ku ∙ v = u ∙ kv • Zero Vector Dot Product Property: 0 ∙ u = 0 • Dot Product & Vector Magnitude Relationship: u ∙ u = |u|

Angle between 2 vectors • If theta is the angle between nonzero vectors a and b, then

Example 2 Find the angle between vectors u and v to the nearest tenth of a degree.

Vector Projection • Let u and v be nonzero vectors and let w 1 and w 2 be vector components of u such that w 1 is parallel to v as shown. • Vector w 1 is called the vector projection of u onto v

Example 3 Find the projection of u onto v. Then write u as the sum of two orthogonal vectors , one of which is the projection of u onto v.