Q93 / 160 · AP EAMCET 2020 (17 Sep Shift 2) · Mathematics

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
    | [a b c ]
  • B
    \(|2[\mathrm{abc}]|\)
  • C
    \(\left|\frac{2[a b c]}{|a \times b+b \times c+c \times a|}\right|\)
  • Correct answer
    D
    \(\left|\frac{[a b c]}{|a \times b+b \times c+c \times a|}\right|\)

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, \(d=\left|\frac{[\mathbf{a}, \mathbf{b}, \mathbf{c}]}{|\mathbf{a} \times \mathbf{b}+\mathbf{b} \times \mathbf{c}+\mathbf{c} \times \mathbf{a}|}\right|\)

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

Relevant Questions