Dot product

Definition

The dot product is a scalar quantity obtained by multiplying the magnitudes of two vectors and the cosine of the angle between them. It provides a measure of how much one vector extends in the direction of another.

5 Must Know Facts For Your Next Test

1. The dot 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 dot product is zero, the vectors are orthogonal (perpendicular).
3. In component form, for vectors \(\mathbf{A} = (A_x, A_y, A_z)\) and \(\mathbf{B} = (B_x, B_y, B_z)\), the dot product is calculated as \(A_x B_x + A_y B_y + A_z B_z\).
4. The dot product can be used to determine if two vectors are parallel; if they are parallel, their dot product will be equal to the product of their magnitudes.
5. It has applications in projecting one vector onto another and in calculating work done when force and displacement are represented as vectors.

Review Questions

• How do you calculate the dot product of two vectors given their components?
• What does it mean if the dot product of two vectors is zero?
• Explain how you would use the dot product to find out if two vectors are parallel.

Related terms

Vector: A quantity that has both magnitude and direction.

Orthogonal Vectors: Two vectors that are perpendicular to each other; their dot product equals zero.

Scalar Projection: \( A_{||B} = |A| \cos(\theta) \), represents how much of vector \(A\) lies in the direction of vector \(B\).