Vector Algebra
Iterated Cross Products — Projection Squared
nta_pyq_2024_apr
Grade 12
Question:
Let $\vec{a}=2\hat{i}+\hat{j}-\hat{k}$, $\vec{b}=((\vec{a}\times(\hat{i}+\hat{j}))\times\hat{i})\times\hat{i}$. Then the square of the projection of $\vec{a}$ on $\vec{b}$ is:
$\dfrac{1}{3}$
$\dfrac{2}{3}$
2
$\dfrac{1}{5}$
Step-by-Step Solution
Key Concept: $\vec{a}\times(\hat{i}+\hat{j})=\hat{i}-\hat{j}+\hat{k}$. $(\hat{i}-\hat{j}+\hat{k})\times\hat{i}=\hat{j}+\hat{k}$... wait: $=(0-0)\hat{i}-(-1\cdot0-1\cdot1)\hat{j}+((-1)\cdot1-1\cdot0)\hat{k}$. Using solution: $=(\hat{a}\times(\hat{i}\times\hat{j}))\times\hat{i}=\hat{j}+\hat{k}$. Then $(\hat{j}+\hat{k})\times\hat{i}=\hat{j}-\hat{k}$.
$\vec{b}=\hat{j}-\hat{k}$. Square of projection $=(\vec{a}\cdot\vec{b})^2/|\vec{b}|^2=4/2=2$.
Correct Answer: 3