Vector Algebra
Dot product using vector cross product identity
nta_pyq_2023_jan
Grade 12
Question:
Let $\vec{u} = \hat{i} - \hat{j} - 2\hat{k}$, $\vec{v} = 2\hat{i} + \hat{j} - \hat{k}$, $\vec{v}\cdot\vec{w} = 2$ and $\vec{v}\times\vec{w} = \vec{u} + \lambda\vec{v}$. Then $\vec{u}\cdot\vec{w}$ is equal to
1
$\frac{3}{2}$
2
$-\frac{2}{3}$
Step-by-Step Solution
Key Concept: Take dot product of both sides of $\vec{v}\times\vec{w} = \vec{u}+\lambda\vec{v}$ with $\vec{w}$ and then with $\vec{v}$ to find $\lambda$ and $\vec{u}\cdot\vec{w}$.
Dot both sides with $\vec{v}$: $0 = \vec{u}\cdot\vec{v} + \lambda|\vec{v}|^2 = (2-1+2)+6\lambda \Rightarrow \lambda=-1/2$. Then $\vec{u}\cdot\vec{w} = -2(-1/2)=1$. Answer: (1)
Correct Answer: 1