Scalar product

Definition

The scalar product, also known as the dot product, is a mathematical operation that takes two vectors and returns a single scalar. It is calculated as the product of the magnitudes of the two vectors and the cosine of the angle between them.

5 Must Know Facts For Your Next Test

1. The scalar product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) is given by \(\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos(\theta)\), where \(\theta\) is the angle between them.
2. If the scalar product of two non-zero vectors is zero, then the vectors are perpendicular (orthogonal).
3. The scalar product can also be computed using component form: \(\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z\) in three-dimensional space.
4. Scalar products are commutative, meaning \(\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}\).
5. The scalar product measures how much one vector extends in the direction of another.

Review Questions

• What is the formula for calculating the scalar product in terms of magnitudes and angle?
• How can you determine if two vectors are perpendicular using their scalar product?
• Is the scalar product commutative? Provide an example.

Related terms

Vector:

A quantity that has both magnitude and direction.

Cross Product:

A vector operation that returns a vector perpendicular to two given vectors in three-dimensional space.

Orthogonal Vectors: \text\\Orthogonal Vectors\\ are vectors that have a dot product equal to zero, indicating they are at right angles to each other.