Vector Algebra
Vector triple product; magnitude of expression
nta_pyq_2023_jan
Grade 12
Question:
Let $\lambda \in \mathbb{R}$, $\vec{a} = \lambda\hat{i}+2\hat{j}-3\hat{k}$, $\vec{b} = \hat{i}-\lambda\hat{j}+2\hat{k}$. If $((\vec{a}+\vec{b})\times(\vec{a}\times\vec{b}))\times(\vec{a}-\vec{b}) = 8\hat{i}-40\hat{j}-24\hat{k}$, then $|\lambda(\vec{a}+\vec{b})\times(\vec{a}-\vec{b})|^2$ is equal to
Step-by-Step Solution
Key Concept: Use the vector identity $(\vec{A}\times\vec{B})\times\vec{C}$ and the given equation to find $\lambda$, then compute the required magnitude.
After simplification using the given vector equation: $\lambda=1$. $(\vec{a}+\vec{b}) = 2\hat{i}+\hat{j}-\hat{k}$, $(\vec{a}-\vec{b}) = 0\hat{i}+3\hat{j}-5\hat{k}$. $(\vec{a}+\vec{b})\times(\vec{a}-\vec{b}) = 2\hat{i}+10\hat{j}+6\hat{k}$. $|\lambda\cdot(\cdot)|^2 = 4+100+36 = 140$. Answer: (1)
Correct Answer: 140