Question

The length of the perpendicular from the origin to the plane passing through three non-collinear points $\mathbf{a},\mathbf{b},\mathbf{c}$ is

• A. $\frac{\lbrack\mathbf{abc}\rbrack}{|\mathbf{a} \times \mathbf{b} + \mathbf{c} \times \mathbf{a} + \mathbf{b} \times \mathbf{c}|}$
• B. $\frac{2\lbrack\mathbf{abc}\rbrack}{|\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}|}$
• C. $\lbrack\mathbf{abc}\rbrack$
• D. None of these

Answer 1

Answer: (A)

$\frac{\lbrack\mathbf{abc}\rbrack}{|\mathbf{a} \times \mathbf{b} + \mathbf{c} \times \mathbf{a} + \mathbf{b} \times \mathbf{c}|}$

Answer 2

The vector equation of the plane passing through points $\mathbf{a},\mathbf{b},\mathbf{c}$ is $\mathbf{r}.(\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}) = \lbrack\mathbf{a}\mspace{6mu}\mathbf{b}\mspace{6mu}\mathbf{c}\rbrack$

Therefore, the length of the perpendicular from the origin to this plane is given by $\frac{\lbrack\mathbf{abc}\rbrack}{|\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}|}$.