Angle Between Vec A and Vec B: Quick Proof & Examples

If vec A – vec B = 0 what is the angle between vec A and vec B

This lesson explains how to determine the angle between vec a and vec b when their dot product equals zero. We give a concise proof, intuitive geometric meaning, quick examples, and a practice question so you can master the concept.

What the problem states

You’re given that vec a · vec b = 0. The task: find the angle between vec a and vec b. This is a classic vector identity that links algebra (dot product) and geometry (angle between vectors).

Short formula reminder

The dot product formula between two non-zero vectors a and b is:

vec a · vec b = |a| |b| cos θ,
where θ is the angle between vec a and vec b.

Why — short proof

Start with vec a · vec b = |a||b| cos θ. Given vec a · vec b = 0 and assuming |a| ≠ 0 and |b| ≠ 0 (non-zero vectors), divide both sides by |a||b| to get cos θ = 0. The only angles in [0, π] whose cosine is zero are θ = π/2, i.e., 90°. So the angle between vec a and vec b is 90°.

Geometric intuition

When two vectors are orthogonal (dot product zero), one has no component in the direction of the other. Graphically, they meet at a right angle. This is why angle between vec a and vec b = 90° is simply the orthogonality condition.

Example

Let vec a = (2, -3, 1) and vec b = (3, 2, 1). Compute the dot product: 2·3 + (−3)·2 + 1·1 = 6 − 6 + 1 = 1. Here the dot product ≠ 0, so the vectors are not orthogonal and the angle between vec a and vec b is not 90°. For truly orthogonal examples, pick vectors like (1,0,0) and (0,1,0) — their dot product is 0 and the angle between vec a and vec b = 90°.

Practice question

If vec a = (a₁,a₂) and vec b = (b₁,b₂) and a₁b₁ + a₂b₂ = 0, what is the angle between vec a and vec b? Answer: 90°.

Want to learn more about the dot product? See this detailed explanation on
Wikipedia — Dot product.
For other lessons on vectors, check our Vectors Basics page.

Key takeaways

• The condition vec a · vec b = 0 (with non-zero vectors) means the vectors are orthogonal.
• The angle between vec a and vec b is 90° (π/2 radians).
• Use the dot product formula |a||b| cos θ to connect algebra to geometry.

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