Length of the perpendicular from the origin to the plane passing through three non-collinear points with…
Length of the perpendicular from the origin to the plane passing through three non-collinear points with position vectors a, \(\mathbf{b}\) and \(\mathbf{c}\) is
| [a b c ]
\(|2[\mathrm{abc}]|\)
\(\left|\frac{2[a b c]}{|a \times b+b \times c+c \times a|}\right|\)
\(\left|\frac{[a b c]}{|a \times b+b \times c+c \times a|}\right|\)
Solution
Vector equation of the plane passing through points with position vector \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) is,
\(\mathbf{r} \cdot(\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a})=[\mathbf{a}, \mathbf{b}, \mathbf{c}]\)
So, length of perpendicular from origin on plane is,
\(d=\left|\frac{[\mathbf{a}, \mathbf{b}, \mathbf{c}]}{|\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}|}\right|\)