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Question: The length of the perpendicular from the origin to the plane passing through the point **a** and con...

The length of the perpendicular from the origin to the plane passing through the point a and containing the line r=b+λc\mathbf{r} = \mathbf{b} + \lambda\mathbf{c} is

A

[abc]∣a×b+b×c+c×a∣\frac{\lbrack\mathbf{abc}\rbrack}{|\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}|}

B

[abc]∣a×b+b×c∣\frac{\lbrack\mathbf{abc}\rbrack}{|\mathbf{a} \times \mathbf{b} + \mathbf{b} \times \mathbf{c}|}

C

[abc]∣b×c+c×a∣\frac{\lbrack\mathbf{abc}\rbrack}{|\mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}|}

D

[abc]∣c×a+a×b∣\frac{\lbrack\mathbf{abc}\rbrack}{|\mathbf{c} \times \mathbf{a} + \mathbf{a} \times \mathbf{b}|}

Answer

[abc]∣b×c+c×a∣\frac{\lbrack\mathbf{abc}\rbrack}{|\mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}|}

Explanation

Solution

The given plane passes through a\mathbf{a} and is parallel to the vectors b−a\mathbf{b} - \mathbf{a} and c\mathbf{c}. So it is normal to (b−a)×c(\mathbf{b} - \mathbf{a}) \times \mathbf{c}. Hence, its equation is (r−a).((b−a)×c)=0(\mathbf{r} - \mathbf{a}).((\mathbf{b} - \mathbf{a}) \times \mathbf{c}) = 0

or r.(b×c+c×a)=[abc]\mathbf{r}.(\mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}) = \lbrack\mathbf{abc}\rbrack

The length of the perpendicular from the origin to this plane is [abc]∣b×c+c×a∣\frac{\lbrack\mathbf{abc}\rbrack}{|\mathbf{b} \times \mathbf{c} + \mathbf{c} \times \mathbf{a}|}.