Testing perpendicular vectors with the dot product

The idea

The dot product of two vectors is the sum of the products of their corresponding components. It also equals the product of their magnitudes and the cosine of their angle. For two nonzero vectors, a zero dot product means a right angle.

Before you begin: Read three-dimensional vector components and calculate a vector magnitude.

Worked example

Show that the following vectors are perpendicular.

a = (1, 2, 2),

b = (2, - 1, 0)

Matching vector components
Vector	x	y	z
a	1	2	2
b	2	−1	0
Products	2	−2	0

Multiply matching components, then add: the three products total zero.

Step 1

Multiply corresponding components and add. The dot product is a scalar, not another vector.

a \cdot b

= 1 (2) + 2 (- 1) + 2 (0)

= 0

Step 2

Check that both vectors have nonzero magnitude before using the angle interpretation.

∣ a ∣ = \sqrt{1^{2} + 2^{2} + 2^{2}}

= 3

∣ b ∣ = \sqrt{2^{2} + {(- 1)}^{2} + 0^{2}}

= \sqrt{5}

Step 3

Since their dot product is zero and both magnitudes are positive, the cosine of their angle is zero.

cos θ = \frac{0}{3 \sqrt{5}} = 0

\Rightarrow θ = 90^{\circ}

A mistake to avoid

Do not infer a three-dimensional angle from how a sketch looks on a screen. Use the components, and remember that a zero vector has no defined direction.

Your turn.

Are these nonzero vectors perpendicular?

u = (1, 0, 2),

v = (2, 3, - 1)

Show the solution

u \cdot v

= 1 (2) + 0 (3) + 2 (- 1)

= 0

Yes. Both vectors are nonzero, and their dot product is zero.

The idea

A mistake to avoid

Your turn.

Put it into practice.

Keep exploring.