Previous Year NEET Question- Introduction To Vectors

• 2019: Let $\vec{a} = 2\hat{i} + 2\hat{j} - \hat{k}, \vec{b} = \hat{i} + 2\hat{j} + \hat{k}, \vec{c} = 3\hat{i} + 4\hat{j} - 2\hat{k}$. Find a vector $\vec{d}$ which is perpendicular to both $\vec{a}$ and $\vec{b}$ and is such that $\vec{d}.\vec{c} = 18$.

Solution:

Let $\vec{d} = x\hat{i} + y\hat{j} + z\hat{k}$.

Since $\vec{d}$ is perpendicular to $\vec{a}$ and $\vec{b}$, we have

$$ \vec{d}\cdot\vec{a} = 0 \implies x + 2y - z = 0 $$

$$ \vec{d} \cdot \vec{b} = 0 \implies x + 2y + z