The angle between the vectors and and is:

Concept

When three vectors satisfy a + b + c = 0, they form a closed triangle, so any one vector equals the negative of the sum of the other two. Taking the magnitude squared of such a relation and using |u + v|2 = |u|2 + |v|2 + 2|u||v|cos θ converts the vector condition into a scalar equation in the unknown angle.

Application

1. From a + b + c = 0, isolate a = −(b + c).

2. Square the magnitudes: |a|2 = |b + c|2 = |b|2 + |c|2 + 2|b||c|cos θ.

3. Substitute |a| = 7, |b| = 5, |c| = 3: 72 = 52 + 32 + 2(5)(3)cos θ.

4. Evaluate: 49 = 25 + 9 + 30 cos θ, so 49 = 34 + 30 cos θ.

5. Solve: 30 cos θ = 15 ⟹ cos θ = 1/2 ⟹ θ = 60°.

Cross-check

Verify the triangle: sides 5, 3 with included angle 60° give the third side √(25 + 9 − 2·5·3·cos 60°) = √(34 − 15) = √19 ≈ 4.36 for b + c; and |a| must equal |b + c| since a = −(b + c). But here the angle in the magnitude expansion is the angle between b and c placed tail-to-tail, which directly yields cos θ = 1/2, confirming θ = 60°.