Let a unit vector $\hat{\mathrm{u}}=x \hat{\mathrm{i}}+y \hat{\mathrm{j}}+z \hat{\mathrm{k}}$ make angles $\frac{\pi}{2}, \frac{\pi}{3}$ and $\frac{2 \pi}{3}$ with the vectors $\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}, \frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}$ and $\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}$ respectively. If $\overrightarrow{\mathrm{v}}=\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}$, then $|\hat{\mathrm{u}}-\overrightarrow{\mathrm{v}}|^{2}$ is equal to

Vector Algebra — JEE Main · 2024 · 29 Jan 2024 · Shift 2 Question with Solution

Let a unit vector

\overset{u}{^} = x \hat{i} + y \hat{j} + z \hat{k}

make angles

\frac{π}{2}, \frac{π}{3}

and

\frac{2 π}{3}

with the vectors

\frac{1}{2} \hat{i} + \frac{1}{2} \hat{k}, \frac{1}{2} \hat{j} + \frac{1}{2} \hat{k}

and

\frac{1}{2} \hat{i} + \frac{1}{2} \hat{j}

respectively. If

v = \frac{1}{2} \hat{i} + \frac{1}{2} \hat{j} + \frac{1}{2} \hat{k}

, then

∣ \overset{u}{^} - v ∣^{2}

is equal to

Answer: (B)

Let

\overset{u}{^} = (x, y, z)

, and let

p_{1} = \frac{1}{2} \hat{i} + \frac{1}{2} \hat{k}, p_{2} = \frac{1}{2} \hat{j} + \frac{1}{2} \hat{k}, p_{3} = \frac{1}{2} \hat{i} + \frac{1}{2} \hat{j}

, each a unit vector.

\overset{u}{^} \cdot p_{1} = cos \frac{π}{2} = 0 \Rightarrow \frac{x + z}{2} = 0 \Rightarrow x + z = 0

\overset{u}{^} \cdot p_{2} = cos \frac{π}{3} = \frac{1}{2} \Rightarrow \frac{y + z}{2} = \frac{1}{2} \Rightarrow y + z = \frac{1}{2}

\overset{u}{^} \cdot p_{3} = cos \frac{2 π}{3} = - \frac{1}{2} \Rightarrow \frac{x + y}{2} = - \frac{1}{2} \Rightarrow x + y = - \frac{1}{2}

.

From

x + z = 0

z = - x

. Substituting in the second equation,

y - x = \frac{1}{2}

. With the third equation

x + y = - \frac{1}{2}

:

Adding:

2 y = 0 \Rightarrow y = 0

, so

x = - \frac{1}{2}

and

z = \frac{1}{2}

\overset{u}{^} = (- \frac{1}{2}, 0, \frac{1}{2})

(check:

∣ \overset{u}{^} ∣^{2} = \frac{1}{2} + 0 + \frac{1}{2} = 1

, valid).

\overset{u}{^} - v = (- \frac{1}{2} - \frac{1}{2}, 0 - \frac{1}{2}, \frac{1}{2} - \frac{1}{2}) = (- 2, - \frac{1}{2}, 0)

∣ \overset{u}{^} - v ∣^{2} = 2 + \frac{1}{2} + 0 = \frac{5}{2}

JEE Main · 2024 · 29 Jan 2024 · Shift 2 · Single correct option