Orthogonal vectors

Definition

Orthogonal vectors are vectors that are perpendicular to each other. Their dot product is zero, indicating no projection of one vector onto the other.

5 Must Know Facts For Your Next Test

1. Two vectors $\mathbf{A}$ and $\mathbf{B}$ are orthogonal if $\mathbf{A} \cdot \mathbf{B} = 0$.
2. Orthogonal vectors form a right angle (90 degrees) with each other.
3. In three-dimensional space, the cross product of two non-parallel vectors results in a vector that is orthogonal to both.
4. Orthogonality is a key concept in vector spaces and is used in applications like projections and coordinate systems.
5. When dealing with basis vectors in Cartesian coordinates, the standard unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$ are mutually orthogonal.

Review Questions

• What condition must be met for two vectors to be considered orthogonal?
• How does the dot product of two orthogonal vectors relate to their magnitudes?
• Can you give an example of three mutually orthogonal unit vectors in three-dimensional space?

Related terms

Dot Product:

An operation that takes two equal-length sequences of numbers and returns a single number. It is calculated as $\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z$ for three-dimensional vectors.

Cross Product:

A binary operation on two vectors in three-dimensional space resulting in another vector perpendicular to the plane containing the original vectors. Given by $\mathbf{A} \times \mathbf{B}$.

Unit Vector:

A vector with a magnitude of one. Often used to indicate direction only without regard to magnitude.