Vector Algebra
Cross Product — Maximum of Scalar Product
nta_pyq_2026_jan
Grade 12
Question:
Let $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\lambda\hat{j}+2\hat{k}$, $\lambda\in\mathbb{Z}$. Let $\vec{c}=\vec{a}\times\vec{b}$ and $\vec{d}$ be a vector of magnitude 2 in $yz$-plane. If $|\vec{c}|=\sqrt{53}$, then the maximum possible value of $(\vec{c}\cdot\vec{d})^2$ is equal to:
Step-by-Step Solution
Key Concept: $\vec{c}=\vec{a}\times\vec{b}=(-2-\lambda)\hat{i}-4\hat{j}+2\lambda\hat{k}$. $|\vec{c}|^2=(2+\lambda)^2+16+4\lambda^2=53\Rightarrow5\lambda^2+4\lambda-33=0\Rightarrow\lambda=-3$ (integer solution).
Maximum of $(\vec{c}\cdot\vec{d})^2=208$.
Correct Answer: 3