Question

If

l_{1}, m_{1}, n_{1}; l_{2}, m_{2}, n_{2}

and

l_{3}, m_{3}, n_{3}

are direction cosines of three mutually perpendicular lines

O A, O B, O C

, show that the line

O P

whose direction cosines are proportional to

l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3}, n_{1} + n_{2} + n_{3}

makes equal angles with lines

O A, O B

and

O C

.

Accepted Answer

Let direction vector of three mutually perpendicular lines be

$\vec{a} = l_{1} \hat{i} + m_{1} \hat{j} + n_{1} \hat{k}$

$\vec{b} = l_{2} \hat{i} + m_{2} \hat{j} + n_{2} \hat{k}$

$\vec{c} = l_{3} \hat{i} + m_{3} \hat{j} + n_{3} \hat{k}$

Let the direction vector associated with direction cosines $l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3}, n_{1} + n_{2} + n_{3}$ be

$\vec{p} = (l_{1} + l_{2} + l_{3}) \hat{i} + (m_{1} + m_{2} + m_{3}) \hat{j} + (n_{1} + n_{2} + n_{3}) \hat{k}$

If two lines with direction ratios $a_{1}, a_{2}, a_{3}$ and $b_{1}, b_{2}, b_{3}$ are perpendicular, then $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$ .

$\Rightarrow l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0$ ...(i)

$\Rightarrow l_{1} l_{3} + m_{1} m_{3} + n_{1} n_{3} = 0$ ...(ii)

$\Rightarrow l_{2} l_{3} + m_{2} m_{3} + n_{2} n_{3} = 0$ ...(iii)

Now, let $x, y, z$ be the angles made by direction vectors $a, b$ and $c$ respectively with $p$

Therefore, $\cos x = \vec{a} \cdot \vec{p}$

$\Rightarrow \cos x = l_{1} (l_{1} + l_{2} + l_{3}) + m_{1} (m_{1} + m_{2} + m_{3}) + n_{1} (n_{1} + n_{2} + n_{3})$

$\Rightarrow \cos x = {l_{1}}^{2} + l_{1} l_{2} + l_{1} l_{3} + m_{1}^{2} + m_{1} m_{2} + m_{1} m_{3} + n_{1}^{2} + n_{1} n_{2} + n_{1} n_{3}$

$\Rightarrow \cos x = {l_{1}}^{2} + m_{1} {}^{2}+ n_{1} {}^{2}+ (l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}) + (l_{1} l_{3} + m_{1} m_{3} + n_{1} n_{3})$

As we know ${l_{1}}^{2} + {m_{1}}^{2} + {n_{1}}^{2} = 1$ [Because sum of squares of direction cosines of a line is equal to $1$ ]

$\Rightarrow \cos x = 1 + 0 = 1$ $[$ from (i) and (ii) $]$

Similarly, $\cos y = 1$ and $\cos z = 1$

$\Rightarrow x = y = z = 0$

Hence, angles made by vector $p$ , with vectors $a, b$ and $c$ are equal.

Important Questions on Three Dimensional Geometry

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If l1, m1, n1;l2, m2, n2 and l3, m3, n3 are direction cosines of three mutually perpendicular lines OA, OB, OC, show that the line OP whose direction cosines are proportional to l1+l2+l3, m1+m2+m3, n1+n2+n3 makes equal angles with lines OA, OB and OC.

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !