Given a =2 i - j + k, b =3 i - k and c =2 i + j -2 k. Find a vector d which is perpendicular to both a and b and c · d =3.

Since $\displaystyle \overrightarrow{\mathbf{d}} \perp \overrightarrow{\mathbf{a}}$ and $\displaystyle \overrightarrow{\mathbf{d}} \perp \overrightarrow{\mathbf{b}} \Rightarrow \overrightarrow{\mathbf{d}}=\lambda(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})$ $$\overrightarrow{\mathbf{d}}=\lambda(\hat{\mathbf{i}}+5 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) $$ $\displaystyle \overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{d}}=3 \Rightarrow 2 \lambda+5 \lambda-6 \lambda=3 \Rightarrow \lambda=3$ $\displaystyle \Rightarrow \overrightarrow{\mathbf{d}}=3 \hat{\mathbf{i}}+\mathbf{1 5} \hat{\mathbf{j}}+\mathbf{9} \hat{\mathbf{k}}$