The equation of the plane in normal form passing through the point $A(\bar{a})$, parallel to a vector $\bar{b}$ and containing a vector $\bar{c}$ is

• A

  $\mathbf{r} \cdot \frac{\mathbf{c} \times \mathbf{a}}{|\mathbf{c} \times \mathbf{a}|}=\left|\frac{\mathbf{a} \times \mathbf{b}}{\mathbf{a} \times \mathbf{c}}\right|$
• B

  $\mathbf{r} \cdot \frac{\mathbf{a} \times \mathbf{b}}{|\mathbf{a} \times \mathbf{b}|}=\frac{[\mathbf{a} \mathbf{b c}]}{|\mathbf{b} \times \mathbf{c}|}$
• Correct answer
  C

  $\mathbf{r} \cdot \frac{\mathbf{b} \times \mathbf{c}}{|\mathbf{b} \times \mathbf{c}|}=\frac{[\mathbf{a} \mathbf{b} \mathbf{c}]}{|\mathbf{b} \times \mathbf{c}|}$
• D

  $\mathbf{r} \cdot[\mathbf{a} \mathbf{b c}] \mathbf{a}=\frac{|\mathbf{b} \times \mathbf{c}|}{|\mathbf{a} \times \mathbf{c}|}$

Solution

$\therefore$ Vector normal to plane $\mathbf{n}=\mathbf{b} \times \mathbf{c}$
$\therefore \quad \mathbf{n}=\frac{\mathbf{b} \times \mathbf{c}}{|\mathbf{b} \times \mathbf{c}|}$ Now Ar $\cdot \mathbf{n}=0$ $(\mathbf{r}-\mathbf{a}) \cdot \frac{(\mathbf{b} \times \mathbf{c})}{|\mathbf{b} \times \mathbf{c}|}=0 \Rightarrow \mathbf{r} \cdot \frac{\mathbf{b} \times \mathbf{c}}{|\mathbf{b} \times \mathbf{c}|}=\mathbf{a} \cdot \frac{(\mathbf{b} \times \mathbf{c})}{|\mathbf{b} \times \mathbf{c}|}$ $\mathbf{r} \cdot \frac{(\mathbf{b} \times \mathbf{c})}{|\mathbf{b} \times \mathbf{c}|}=\frac{[\mathbf{a} \mathbf{b} \mathbf{c}]}{|\mathbf{b} \times \mathbf{c}|}$

Hence, the correct answer is option: 'C'.