Find the length of the vector represented by the directed line segment with initial point P(2, -3, 4) and terminal point Q(-2, 1, 1).

Question

Question

This question was previously asked in

AAI ATC Junior Executive 27 July 2022 Shift 1 Official Paper

Answer 1

Concept-

Length of vector \(a\hat{i}+b\hat{j}+c\hat{k}= \sqrt{(a)^{2}+(b)^{2}+(c)^{2}} \)

Let Initial point A(x₁, y₁, z₁) and terminal point B(x₂, y2, z2) of a vector than

\(|\vec{AB}|=(x_{2}- x_{1})\hat{i}+(y_{2}- y_{1})\hat{i}+(z_{2}- z_{1})\hat{i} \)

Calculation-

Initial point is P(2,-3,4) and terminal point Q(-2, 1, 1)

Vector \(|\vec{PQ}|\) = (-2 - 2)\(\hat{i}\) + (1 - (-3))\(\hat{j}\)+ (1 - 4)\(\hat{k}\)

Vector \(|\vec{PQ}| \) = -4\(\hat{i}\) + 4\(\hat{j}\) - 3\(\hat{k}\)

Now the length of a vector \(|\vec{PQ}| \) is given by

\(|\vec{PQ}| =\sqrt{(-4)^2+(4)^2+(-3)^2}\)

\(|\vec{PQ}| =\sqrt{16+16+9}\)

\(|\vec{PQ}|= \sqrt{41} \)

∴ length of vector \(|\vec{PQ}| ~is ~ \sqrt{41}\)