Let $\vec{d}$ = $x\hat{i}+y\hat{j}+z\hat{k}$ since $\vec{d}$ is perpendicular to $\vec{c}$ and $\vec{b}$ , so their dot product is zero $\vec{d}\cdot\vec{c}$ = 0 and $\vec{d}\cdot\vec{b}$ = 0 3x + y - z = 0 ... (1) x - 4y + 5z = 0 ... (2) Also given that $\vec{d}\cdot\vec{a}$ = 21. 4x + 5y - z = 21 ... (3) Solving all the equations simultaneously x = $-\frac{1}{3}$ , y = $\frac{16}{3}$ , z = $\frac{13}{3}$ $\vec{d}$ = $\frac{1}{3}$ $(-\hat{i}+16\hat{j}+13\hat{k})$