Vector Algebra
Vector Algebra
nta_abhyas_2025
Grade 12
Question:
Let $\vec{U} = \vec{i} + \vec{j}$, $\vec{V} = \vec{i} - \vec{j}$ and $\vec{W} = 3\vec{i} + 5\vec{j} + 3\vec{k}$. If $\vec{n}$ is a unit vector such that $\vec{U} \cdot \vec{n} = 0$ and $\vec{V} \cdot \vec{n} = 0$, then $|\vec{W} \cdot \vec{n}|$ is equal to
Step-by-Step Solution
Key Concept: Cross product of two vectors yields a perpendicular vector; normalizing using unit vector condition
$\vec{n}$ is perpendicular to both $\vec{U}$ and $\vec{V}$, so $\vec{n} = \vec{U} \times \vec{V}$. Computing the cross product gives $\vec{n} = 3\vec{a}(-2\vec{b})$. Since $\vec{n}$ is a unit vector, $|\vec{n}| = 1$, so $2\lambda = 1$ and $\lambda = \frac{1}{2}$. Therefore $\vec{n} = \frac{1}{2}\vec{i} + \vec{k}$ and $\vec{W} \cdot \vec{n} = 3$.
Correct Answer: 3