Vector Algebra
Unit Vector — Two Angle Conditions
nta_pyq_2024_apr
Grade 12
Question:
Let a unit vector which makes an angle of $60^\circ$ with $2\hat{i}+2\hat{j}-\hat{k}$ and angle $45^\circ$ with $\hat{i}-\hat{k}$ be $\overrightarrow{C}$. Then $\overrightarrow{C}+\left(-\dfrac{1}{2}\hat{i}+\dfrac{1}{3\sqrt{2}}\hat{j}-\dfrac{\sqrt{2}}{3}\hat{k}\right)$ is:
$\dfrac{\sqrt{2}}{3}\hat{i}-\dfrac{1}{2}\hat{k}$
$\left(\dfrac{1}{\sqrt{3}}+\dfrac{1}{2}\right)\hat{i}+\left(\dfrac{1}{\sqrt{3}}-\dfrac{1}{3\sqrt{2}}\right)\hat{j}+\left(\dfrac{1}{\sqrt{3}}+\dfrac{\sqrt{2}}{3}\right)\hat{k}$
$\dfrac{\sqrt{2}}{3}\hat{i}+\dfrac{1}{3\sqrt{2}}\hat{j}-\dfrac{1}{2}\hat{k}$
$-\dfrac{\sqrt{2}}{3}\hat{i}+\dfrac{\sqrt{2}}{3}\hat{j}+\left(\dfrac{1}{2}+\dfrac{2\sqrt{2}}{3}\right)\hat{k}$
Step-by-Step Solution
Key Concept: Let $\overrightarrow{C}=C_1\hat{i}+C_2\hat{j}+C_3\hat{k}$, $C_1^2+C_2^2+C_3^2=1$. Condition 1: $\overrightarrow{C}\cdot(2\hat{i}+2\hat{j}-\hat{k})=3\cos60^\circ=3/2\Rightarrow2C_1+2C_2-C_3=3/2$. Condition 2: $\overrightarrow{C}\cdot(\hat{i}-\hat{k})=\sqrt{2}\cos45^\circ=1\Rightarrow C_1-C_3=1$.
$C_1=\sqrt{2}/3+1/2$, $C_2=-1/(3\sqrt{2})$, $C_3=\sqrt{2}/3-1/2$. Sum $=\frac{\sqrt{2}}{3}\hat{i}-\frac{1}{2}\hat{k}$.
Correct Answer: 1