Dot Product

The dot product or scalar product of two vectors is the product of their magnitudes multiplied by the cosine of the angle that they form.

Dot Product

It can also be expressed as:

Dot Product

Example

Find the dot product of two vectors whose coordinates in an orthonormal basis are: (1, 1/2, 3) and (4, −4, 1).

(1, 1/2, 3) · (4, −4, 1) = 1 · 4 + (1/2) · (−4) + 3 · 1 = 4 −2 + 3 = 5


Magnitude of a Vector

Magnitude of a Vector

Example

Find the magnitude of a vector with coordinates Vector U = (−3, 2, 5) in an orthonormal basis.

Dot Product Example


Angle between Two Vectors

Angle Between Two Vectors

Example

Determine the angle between the vectors vector u = (1, 2, −3) and = (−2, 4, 1).

Dot Product Example

Dot Product Example


Orthogonal Vectors

Two vectors are orthogonal or perpendicular if their dot product is zero.

Orthogonal Vectors

Example

Calculate the x and y values for the vector (x, y, 1) that is orthogonal to the vectors (3, 2, 0) and (2, 1, −1).

Orthogonal Vector Example

Orthogonal Vector Example

Orthogonal Vector Solution

Properties of the Dot Product

1Commutative

Commutative Properties

2 Associative

Associative Properties

3 Distributive

Distributive Properties

4

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Geometric Interpretation of the Dot Product

Scalar Projection

Scalar Projection

Scalar Projection

Scalar Projection

Vector Projection

The vector projection is the unit vector of by the scalar projection of u on v.

The scalar projection of u on v is the magnitude of the vector projection of u on v.

Exercise

Given the vectors Vector U and Vector V, calculate:

1. The magnitudes of vector u and ·

Magnitude of Vectors

Magnitude of Vectors

2. The dot product of Vector U and ·

Dot Product Example

3. The angle.

Dot Product Angle

Dot Product Angle

4. The value of m for which vectors Vector U and Vector are orthogonal.

Orthogonal Vector Example