P and Q are the points (3, −1, 2) and (1, 4, −2) respectively. The vector components of \(\overrightarrow{PQ}\) along the x-axis, y-axis and z-axis are
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(-2\hat{\imath}\), \(5\hat{\jmath}\) and \(-4\hat{k}\)
The vector from P to Q is (end point) minus (starting point):
\(\overrightarrow{PQ} = (1-3)\hat{\imath}+(4+1)\hat{\jmath}+(-2-2)\hat{k}\)
\(\overrightarrow{PQ} = -2\hat{\imath}+5\hat{\jmath}-4\hat{k}\)
The scalar components are the numbers \(-2, 5, -4\). The vector components are the three vectors \(-2\hat{\imath}\), \(5\hat{\jmath}\), \(-4\hat{k}\), whose sum is \(\overrightarrow{PQ}\).
Why the other options are incorrect
Option (1) \(-2\), \(5\) and \(-4\) — Incorrect. These are the scalar components, which are numbers, not vectors.
Option (3) \(2\hat{\imath}\), \(-5\hat{\jmath}\) and \(4\hat{k}\) — Incorrect. These are the vector components of \(\overrightarrow{QP}\), the vector from Q to P.
Option (4) \(4\hat{\imath}\), \(3\hat{\jmath}\) and \(0\hat{k}\) — Incorrect. These come from adding the coordinates of P and Q instead of subtracting them.