Perpendicular Vector — Definition, Formula & Examples

A perpendicular vector is a vector that forms a 90° angle with another given vector. Two vectors are perpendicular (also called orthogonal) when their dot product equals zero.

Two vectors $\mathbf{a}$ and $\mathbf{b}$ in $\mathbb{R}^n$ are perpendicular if and only if $\mathbf{a} \cdot \mathbf{b} = 0$ , where $\mathbf{a} \cdot \mathbf{b}$ denotes the dot product. A perpendicular vector to $\mathbf{a}$ is any nonzero vector $\mathbf{b}$ satisfying this condition.

Key Formula

\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + \cdots + a_n b_n = 0

Where:

$\mathbf{a}, \mathbf{b}$ = Two vectors in \mathbb{R}^n
$a_i, b_i$ = The corresponding components of each vector

How It Works

To check whether two vectors are perpendicular, compute their dot product. If the result is zero, the vectors are perpendicular. To find a perpendicular vector in 2D, you can swap the components of the original vector and negate one of them: if

\mathbf{a} = (a_1, a_2)

, then

(-a_2, a_1)

is perpendicular to

\mathbf{a}

. In 3D, the cross product

\mathbf{a} \times \mathbf{b}

produces a vector perpendicular to both

\mathbf{a}

and

\mathbf{b}

Worked Example

Problem: Determine whether

\mathbf{a} = (3, -2, 1)

and

\mathbf{b} = (1, 1, -1)

are perpendicular.

Compute the dot product: Multiply corresponding components and add the results.

\mathbf{a} \cdot \mathbf{b} = (3)(1) + (-2)(1) + (1)(-1) = 3 - 2 - 1 = 0

Interpret the result: Since the dot product equals zero, the two vectors are perpendicular.

Answer:

\mathbf{a}

and

\mathbf{b}

are perpendicular because

\mathbf{a} \cdot \mathbf{b} = 0

Why It Matters

Perpendicular vectors are central to constructing coordinate systems, defining normal directions to surfaces, and projecting one vector onto another. In physics and engineering, normal vectors determine how light reflects off surfaces and how forces decompose into independent components. Linear algebra courses rely heavily on orthogonality when building orthonormal bases and performing Gram-Schmidt orthogonalization.

Common Mistakes

Mistake: Assuming two vectors must have the same magnitude to be perpendicular.

Correction: Perpendicularity depends only on direction, not magnitude. Vectors of any length can be perpendicular as long as their dot product is zero.