Scalar Product of Vectors

The scalar product or dot product of two vectors Vector and Vector is equal to:

Scalar Product

Example

Scalar Product Example

Scalar Product Calculations

Scalar Product Solution


It can also be expressed as:

Scalar Product Expression

Example

Scalar Product Example

Scalar Product Solution


Magnitude of a Vector

Magnitude of a Vector Formula

Examples

Scalar Product Example

Scalar Product Calculations

Scalar Product Solution

Properties of the Scalar Product

1Commutative

Commutative Properties of the Scalar Product

2 Associative

Associative Properties of the Scalar Product

3 Distributive

Distributive Properties of the Scalar Product

4

The dot product of a non-zero vector is always positive

Properties of the Scalar Product


Examples

Calculate the scalar product of the following vectors:

1. Vector = (3, 4) and =(-8, 6)

Vector · = 3 · (-8) + 4 · 6 = 0

2. Vector = (5, 6) and =(-1, 4)

Vector · = 5 · (-1) + 6 · 4 = 19

3. Vector = (3, 5) and =(-1, 6)

Vector · = 3 · (-1) + 5 · 6 = 27


If B = {Vector , Vector} is a basis of vectors in the plane, such that |Vector | = |Vector| = 2 and cos (Vector, ) = 1/2 and:

Vector = 3Vector + 2Vector and Vector = Vector + 2.

Calculate Vector · Vector.

Scalar Product Operations

Scalar Product Calculations

The scalar product is commutative.

Commutative Scalar Product

cos(Vector, Vector) = cos (Vector, ) = 1

Scalar Product Calculations

Scalar Product Solution