Vector Algebra
Rotation of vector through right angle; projection
nta_pyq_2023_jan
Grade 12
Question:
The vector $\vec{a} = -\hat{i} + 2\hat{j} + \hat{k}$ is rotated through a right angle, passing through the y-axis in its way and the resulting vector is $\vec{b}$. Then the projection of $3\vec{a} + \sqrt{2}\vec{b}$ on $\vec{c} = 5\hat{i} + 4\hat{j} + 3\hat{k}$ is
Step-by-Step Solution
Key Concept: Find $\vec{b}$ by rotating $\vec{a}$ 90° in the plane containing $\vec{a}$ and the y-axis, maintaining unit magnitude. Then compute projection.
$\vec{b} = -\sqrt{2}(-\hat{i}-\hat{j}+\hat{k})$. $3\vec{a}+\sqrt{2}\vec{b} = (-3+2)\hat{i}+(6+2)\hat{j}+(3-2)\hat{k} = (-1,8,1)$... projection $= \frac{(-1)(5)+(8)(4)+(1)(3)}{\sqrt{50}} = \frac{30}{5\sqrt{2}} = 3\sqrt{2}$. Answer: (1)
Correct Answer: $3\sqrt{2}$