Let a unit vector which makes an angle of 60 ° with 2 hat i +2 hat j -hat k and an angle of 45 ° with hat i -hat k be vec C . Then overrightarrow C + (-1/2 hat i +frac 1 3 √(2) hat j -frac √(2) 3 hat k ) is :

-frac √(2) 3 hat i +frac √(2) 3 hat j + (1/2+frac 2 √(2) 3 ) hat k

frac √(2) 3 hat i +frac 1 3 √(2) hat j -1/2 hat k

(frac 1 √(3) +1/2 ) hat i + (frac 1 √(3) -frac 1 3 √(2) ) hat j + (frac 1 √(3) +frac √(2) 3 ) hat k

Mathematics · Vector Algebra

JEE Main 2024 — 4 April, Shift 1 — Question 11

Let a unit vector which makes an angle of $60^{\circ}$ with $2 \hat{i}+2 \hat{j}-\hat{k}$ and an angle of $45^{\circ}$ with $\hat{i}-\hat{k}$ be $\vec{C}$ . Then $\overrightarrow{\mathrm{C}}+\left(-\frac{1}{2} \hat{\mathrm{i}}+\frac{1}{3 \sqrt{2}} \hat{\mathrm{j}}-\frac{\sqrt{2}}{3} \hat{\mathrm{k}}\right)$ is :

Option A:
$-\frac{\sqrt{2}}{3} \hat{\mathrm{i}}+\frac{\sqrt{2}}{3} \hat{\mathrm{j}}+\left(\frac{1}{2}+\frac{2 \sqrt{2}}{3}\right) \hat{\mathrm{k}}$
Option B:
$\frac{\sqrt{2}}{3} \hat{i}+\frac{1}{3 \sqrt{2}} \hat{j}-\frac{1}{2} \hat{k}$
Option C:
$\left(\frac{1}{\sqrt{3}}+\frac{1}{2}\right) \hat{\mathrm{i}}+\left(\frac{1}{\sqrt{3}}-\frac{1}{3 \sqrt{2}}\right) \hat{\mathrm{j}}+\left(\frac{1}{\sqrt{3}}+\frac{\sqrt{2}}{3}\right) \hat{\mathrm{k}}$
Option D:
$\frac{\sqrt{2}}{3} \hat{i}-\frac{1}{2} \hat{k}$
Correct

Answer: D

Step-by-step solution

$\overrightarrow{\mathrm{C}}=\mathrm{C}_{1} \hat{i}+\mathrm{C}_{2} \hat{\mathrm{j}}+\mathrm{C}_{3} \hat{k}$

$\mathrm{C}_{1}^{2}+\mathrm{C}_{2}^{2}+\mathrm{C}_{3}^{2}=1$

$\overrightarrow{\mathrm{C}} \cdot(2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}})=|C| \sqrt{9} \cos 60^{\circ}$

$2 \mathrm{C}_{1}+2 \mathrm{C}_{2}-\mathrm{C}_{3}=\frac{3}{2}$

$\mathrm{C}_{1}-\mathrm{C}_{3}=1$

$\mathrm{C}_{1}+2 \mathrm{C}_{2}=\frac{1}{2}$

$\mathrm{C}_{1}=\frac{\sqrt{2}}{3}+\frac{1}{2}$

$\mathrm{C}_{2}=\frac{-1}{3 \sqrt{2}}$

$\mathrm{C}_{3}=\frac{\sqrt{2}}{3}-\frac{1}{2}$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 4 April, Shift 1
Subject: Mathematics
Chapter: Vector Algebra
Topic: Scalar or Dot Product of Two Vectors

Let ABC be a triangle of area 15√(2) . Let the vectors representing its sides be: overrightarrowAB = hati + 2hatj - 7hatk overrightarrowBC…

Answer: D

Step-by-step solution

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