Name: Distance of a Point From a Plane
Uploaded: 2019-07-19
Duration: 9 min 54 s
Description: Three-dimensional space has an infinite number of points and planes in it! This video discusses the distance of a point from a plane, both in vector and cart...

Question

The fig. shows a rectangular box of size a, b and c units along x, y and z-axes respectively. Prove that the angles between the four diagonals are given by $\cos^{- 1} (\frac{\pm a^{2} \pm b^{2} \pm c^{2}}{a^{2} + b^{2} + c^{2}})$

where all signs cannot be +ve or -ve.

Accepted Answer

The co-ordinates of the vertices of the rectangular box are $O (0, 0, 0), A (a, 0, 0)$ $B (0, b, 0) C (0, 0, c) P (a, b, c) M (a, b, 0) N (0, b, c) L (a, 0, c)$ .

Clearly $O P, A N, C M & B L$ are four diagonal of the rectangular box.

Therefore, direction ratios of the diagonals $O P, A N, B L, C M$ are proportional to

$a, b, c, - a, b, c, a, - b, c & a, b, - c$ respectively

Let $θ_{1}$ the angle between $O P & A N$ then

$\cos θ_{1} = \frac{a (- a) + (b) (b) + (c) (c)}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{(- a)^{2} + b^{2} + c^{2}}} = \frac{- a^{2} + b^{2} + c^{2}}{a^{2} + b^{2} + c^{2}}$

$θ_{1} = \cos^{- 1} (\frac{- a^{2} + b^{2} + c^{2}}{a^{2} + b^{2} + c^{2}})$

Now let $θ_{2}$ be the angle between $O P & B L$ then

$\cos θ_{2} = \frac{a \times a + b \times - b + c \times c}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{a^{2} + (- b)^{2} + c^{2}}} = \frac{a^{2} - b^{2} + c^{2}}{a^{2} + b^{2} + c^{2}}$

$θ_{2} = \cos^{- 1} (\frac{a^{2} - b^{2} + c^{2}}{a^{2} + b^{2} + c^{2}})$

Let $θ_{3}$ be the angle between diagonal $C M & B L$

$\cos θ_{3} = \frac{a \times a + (- b) \times b + c \times - c}{\sqrt{a^{2} + (- b)^{2} + c^{2}} \sqrt{a^{2} + b^{2} + (c)^{2}}} = \frac{a^{2} - b^{2} - c^{2}}{a^{2} + b^{2} + c^{2}}$

$θ_{3} = \cos^{- 1} (\frac{a^{2} - b^{2} - c^{2}}{a^{2} + b^{2} + c^{2}})$

Similarly, the angle between the other pairs of diagonals can be obtained.

Putting all the results together, we obtain that the angles between the diagonals of the rectangular box are given by $\cos^{- 1} (\frac{a^{2} \pm b^{2} \pm c^{2}}{a^{2} + b^{2} + c^{2}})$ .

The fig. shows a rectangular box of size a, b and c units along x, y and z-axes respectively. Prove that the angles between the four diagonals are given by $\cos^{- 1} (\frac{\pm a^{2} \pm b^{2} \pm c^{2}}{a^{2} + b^{2} + c^{2}})$

where all signs cannot be +ve or -ve.

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

The fig. shows a rectangular box of size a, b and c units along x, y and z-axes respectively. Prove that the angles between the four diagonals are given by cos-1±a2±b2±c2a2+b2+c2 where all signs cannot be +ve or -ve.

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

The fig. shows a rectangular box of size a, b and c units along x, y and z-axes respectively. Prove that the angles between the four diagonals are given by $\cos^{- 1} (\frac{\pm a^{2} \pm b^{2} \pm c^{2}}{a^{2} + b^{2} + c^{2}})$

where all signs cannot be +ve or -ve.