Q.If $\vec{a} = \vec{b} + \vec{c}$, then is it true that $|\vec{a}| = |\vec{b}| + |\vec{c}|$? Justify your answer.

Concept: Triangle Inequality — the magnitude of a sum of vectors is at most the sum of their magnitudes, with equality only when the vectors are parallel and in the same direction.

1. Given $\vec{a} = \vec{b} + \vec{c}$, we take magnitudes on both sides: $|\vec{a}| = |\vec{b} + \vec{c}|$.
2. By the triangle inequality, $|\vec{b} + \vec{c}| \leq |\vec{b}| + |\vec{c}|$, with equality if and only if $\vec{b}$ and $\vec{c}$ are in the same direction. …

The question asks whether vector addition implies scalar addition of magnitudes. This is a classic trap — it confuses the vector sum with the scalar sum of lengths.

Why the Triangle Inequality is the key

When you add two vectors $\vec{b}$ and $\vec{c}$, the result $\vec{a}$ is the third side of a triangle formed by placing $\vec{b}$ and $\vec{c}$ head-to-tail. The length of that third side depends on the angle between $\vec{b}$ and $\vec{c}$.

Think of it physically: if you walk 3 km east, then 4 km north, you end up 5 km from the start — not 7 km. The path matters. The magnitude of the sum is at most the sum of the magnitudes, and equals it only when both vectors point exactly the same way.

Let's prove this step by step.

1. Start with the square of the magnitude. For any vector $\vec{a} = \vec{b} + \vec{c}$, we have:

Expanding the dot product:

where $\theta$ is the angle between $\vec{b}$ and $\vec{c}$.

2. Compare with $(|\vec{b}| + |\vec{c}|)^2$. Expand the scalar sum:

The only difference is the factor $\cos\theta$ in the cross term.

3. Apply the range of cosine. Since $-1 \leq \cos\theta \leq 1$, we have:

Therefore:

Taking square roots (all quantities are non-negative):

4. When does equality occur? …

Method: Testing a 'Magnitude of a Sum' Claim with the Triangle Inequality

This is a reasoning question: decide whether $|\vec b+\vec c|$ equals, or only bounds, $|\vec b|+|\vec c|$.

Steps

Step 1: Invoke the triangle inequality

For any two vectors, $|\vec b+\vec c|\le|\vec b|+|\vec c|$. So equality is the special case, not the rule — which already tells you the claim is not always true.

Step 2: Find the equality condition by squaring

Expand $|\vec b+\vec c|^2=|\vec b|^2+|\vec c|^2+2|\vec b||\vec c|\cos\theta$ and compare with $(|\vec b|+|\vec c|)^2=|\vec b|^2+|\vec c|^2+2|\vec b||\vec c|$. They match only when $\cos\theta=1$. …

Common Mistakes

Mistake 1: Claiming $|\vec b+\vec c|=|\vec b|+|\vec c|$ always holds

Why it's wrong: this is only the equality case of the triangle inequality, valid when the vectors are parallel and same-sense. Correct approach: in general $|\vec b+\vec c|\le|\vec b|+|\vec c|$, strict unless collinear and codirectional.

Mistake 2: Giving a bare 'no' with no justification or condition

Why it's wrong: the question says 'justify', and the interesting content is when equality holds. Correct approach: state the equality condition ($\theta=0$) and back it with a counterexample. …