Question

Let \[ \vec{a}=\hat{i}+x\hat{j}+\hat{k}, \qquad \vec{b}=\hat{i}+\hat{j}+\hat{k} \] and \[ |\vec{a}+\vec{b}|=|\vec{a}|+|\vec{b}|. \] Then

• A. \(x=1\)
• B. \(x=-1\)
• C. \(x=0\)
• D. No such real \(x\) exists

Hint:

The equality \[ |\vec{a}+\vec{b}|=|\vec{a}|+|\vec{b}| \] holds only when the vectors are parallel and point in the same direction.

Answer 1

The Correct Option is A

Step 1: Recall when equality holds in the triangle inequality.
\(|\vec{a}+\vec{b}| = |\vec{a}|+|\vec{b}|\) iff \(\vec{a}\) and \(\vec{b}\) point in the same direction, i.e., \(\vec{a} = \lambda\vec{b}\) for some \(\lambda > 0\).

Step 2: Find \(x\).
\(\vec{a} = (1,x,1)\), \(\vec{b}=(1,1,1)\). For \(\vec{a}=\lambda\vec{b}\): from the first component \(\lambda=1\), so \(x=1\).
\[\boxed{x = 1}\]