1. Sec 13.3The Dot Product Definition: The dot product is sometimes called the scalar product or the inner product of two vectors.

2. Properties of the Dot Product If a, b, and c are vectors, and k is a scalar, then

3. Theorem: If θ is the angle between the vectors a and b, thena ∙ b = |a||b| cos θ Corollary: If θ is the angle between the non-zero vectors a and b, then Definition: Perpendicular vectors are also called orthogonal vectors. Corollary: Two vectors a and b are orthogonal if and only if a ∙ b = 0

4. Direction Angles and Direction Cosines The direction angles of a non-zero vector a are the angles α, β, and γ (in the interval [0, π]) that a makes with the positive x-, y-, and z-axes. The cosines of these direction angles: cos α, cos β, and cos γ are called the direction cosines of the vector a. Theorem:

5. Vector and Scalar Projections The vector projection of b onto a is denoted and defined by The scalar projection of b onto a (also called the component of b along a) is defined to be the signed magnitude of the vector projection, which is the number |b| cos θ, where θ is the angle between a and b.

6. Sec 13.4The Cross Product Definition: Note: The cross product is defined only for three-dimensional vectors.

7. Theorem: The vector a × b is orthogonal to both a and b. Theorem: If θ is the angle between a and b (so 0 ≤ θ ≤ π), then |a × b| = |a| |b| sin θ Corollary: Two non-zero vectors a and b are parallel if and only if a × b = 0 Theorem: The length of the cross product a × b is equal to the area of the parallelogram determined by a and b.

8. Theorem: i × j = k j × k = i k × i = j j × i = −k k × j = − i i × k = −j Theorem: If a, b, and c are vectors and k is a scalar, then 1.a × b = − b × a 2.(ka) × b = k(a × b) = a × (kb) 3. a × (b + c) = a × b + a × c 4.(a + b) × c = a × c + b × c 5.a ∙ (b × c) = (a × b) ∙ c 6.a × (b × c) = (a ∙ c) b − (a ∙ b) c Note: The cross product is neither commutative nor associative.

9. Triple Products Definition: Theorem: The volume of the parallelepiped determined by the vectors a, b, and c is the magnitude of their scalar product: V = |a ∙ (b × c)|