Sec 13.3 The Dot Product

Definition:

The dot product is sometimes called the scalar product or the inner product of two vectors.

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Properties of the Dot Product

If a, b, and c are vectors, and k is a scalar, then

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Theorem: If θ is the angle between the vectors a and b, then a ∙ 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

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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:

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cos , cos , cos

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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.

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Sec 13.4 The Cross Product

Definition:

Note:

The cross product is defined only for three-dimensional vectors.

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then the,, and ,, If

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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.

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, then1. a × b = − b × a2. (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) ∙ c6. a × (b × c) = (a ∙ c) b − (a ∙ b) c

Note: The cross product is neither commutative nor associative.

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)|

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