How can I use a dot product to decide whether two vectors are perpendicular?
Quick answer
Multiply matching components of the two vectors, then add the products. For two nonzero vectors, a dot product of zero means they are perpendicular; a nonzero result means the test for perpendicularity is not met. For example, (1, 2)·(2, −1) = 1×2 + 2×(−1) = 0, so that pair is perpendicular. [S1]
The question, the details, the next step.
How do I use a dot product to check whether two vectors are perpendicular? I have their components, but I’m not sure what to calculate.
For two-dimensional vectors, multiply their matching components and add those products. If the vectors are u = (u₁, u₂) and v = (v₁, v₂), calculate u·v = u₁v₁ + u₂v₂. For two nonzero vectors, the result is zero if and only if they are perpendicular, also called orthogonal. [S1]
So the practical test is: calculate the dot product; if it equals zero, the nonzero vectors are perpendicular. If it does not equal zero, they are not perpendicular. Keep the nonzero condition in mind: the test’s stated conclusion applies to two nonzero vectors. [S1]
Can you show the calculation with a pair that is perpendicular, including how you keep track of the signs?
Take u = (1, 2) and v = (2, −1). Match the first components, then the second components:
- First pair: 1×2 = 2.
- Second pair: 2×(−1) = −2.
- Add: 2 + (−2) = 0.
Therefore u·v = 0, so these two nonzero vectors are perpendicular. The negative product matters: it cancels the positive product. OpenStax gives this same pair as an example of the zero test. [S1]
A quick check is to write both multiplication steps before adding. That makes it easier to notice a missed negative sign or a component paired with the wrong component.
What would a nonzero result look like, and does it tell me the angle between the vectors?
Try u = (1, 2) and w = (1, 1):
- Multiply matching components: 1×1 = 1 and 2×1 = 2.
- Add: 1 + 2 = 3.
Since the dot product is 3 rather than zero, the perpendicularity test is not met; this pair is not perpendicular. [S1]
A nonzero result by itself does not tell you the exact angle. The dot product relates to angle using u·v = ||u||||v|| cos θ, and the zero result for nonzero vectors establishes a 90° angle. This test answers whether the angle is 90°, not what every other angle is. [S1]
Can you give me a reusable way to show my work, and what should I do if one vector is zero?
Copy and fill in this template:
u = ([a], [b]), v = ([c], [d]) u·v = ([a]×[c]) + ([b]×[d]) = [result] If both vectors are nonzero and [result] = 0, they are perpendicular. If both are nonzero and [result] ≠ 0, they are not perpendicular.
For example, with (1, 2) and (2, −1), the calculation is (1×2) + (2×−1) = 2 − 2 = 0, so the pair is perpendicular. The theorem behind this conclusion is stated for two nonzero vectors. If one vector is zero, do not use that theorem to conclude the pair is perpendicular; check the definition or convention used in your class instead. [S1]
The formula and nonzero-vector test establish the result for two nonzero vectors. For a pair involving the zero vector, follow your course’s definition.
A practical checklist
Write both vectors in component form and check that their components correspond in the same order.
Multiply matching components, keeping negative signs.
Add the products to get the dot product.
For two nonzero vectors, conclude they are perpendicular exactly when the result is zero.
If a vector is zero, check the definition or convention your class uses before concluding.
A few more useful answers
Do I multiply the components in the same positions?
Yes. For two-dimensional vectors, multiply first component by first component and second by second, then add: u·v = u₁v₁ + u₂v₂. [S1]
If the dot product is not zero, are the vectors perpendicular?
No. For two nonzero vectors, a nonzero dot product means the zero test for perpendicularity is not met. [S1]
Does a zero dot product tell me the vectors have the same length?
No. A zero dot product establishes perpendicularity for two nonzero vectors; it does not establish that their lengths are equal. [S1]
Sources
References used for this guide.
- [S1] OpenStax Calculus Volume 3: The Dot Product openstax.org · Checked