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  • . If l1,m1,n1,l2,m2,n2,l3,m3,n3 are the direction cosines of 3 mutually perpendicular lines, prove that the line whose direction cosines are proportional to l1 plus m1 plus n1 plus l2 plus m2 plus n2 plus l3 plus m3 plus n3 makes equal angles with them.

If l1, m1, n1; l2, m2, n2; l3, m3, n3 are the direction cosines of 3 mutually perpendicular lines, prove that the line whose direction cosines are proportional to l1 + l2 + l3 , m1 + m2 + m3 , n1 + n2 + n3 makes equal angles with them.

Answers (1)

Let the direction vector of the 3 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 vectors 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}=\left (l_{1} + l_{2} + l_{3} \right )\hat{i}+\left (m_{1} + m_{2} + m_{3} \right )\hat{j}+\left ( n_{1} + n_{2} + n_{3} \right )\hat{k}

Since the lines associated with the direction vectors a, b, and c are mutually perpendicular, we get

\vec{a}.\vec{b}=0 (Since the dot product of two perpendicular vectors is 0)

=>l_{1}l_{2} + m_{1}m_{2}+n_{1}n_{2}=0 …(1)

Similarly, 

l_{1}l_{3} + m_{1}m_{3}+n_{1}n_{3}=0 …(2)

Finally, \vec{b}.\vec{c}=0

=>l_{2}l_{3} + m_{2}m_{3}+n_{2}n_{3}=0 …(3)

Now, let us consider x, y and z as the angles made by direction vectors a, b, and c respectively with p.

Then,

\cos x=\vec{a}.\vec{p}
\Rightarrow \cos x =l_{1}\left ( l_{1}+l_{2}+l_{3} \right )+m_{1}\left ( m_{1}+m_{2}+m_{3} \right )+n_{1}\left ( n_{1}+n_{2}+n_{3} \right )\\ \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}+\left ( l_{1}l_{2}+m_{1}m_{2}+ n_{1}n_{2} \right )+\left ( l_{1}l_{3}+m_{1}m_{3}+ n_{1}n_{3} \right )\\

We know, l_{1}^{2}+m_{1}^{2}+n_{1}^{2}=1  [since the sum of squares of direction cosines of a line = 1]

\Rightarrow \cos x=1+0=1 [from (1) and (2)]

Then, \cos y=1 and \cos z=1

=> x = y = z = 0.

Therefore, the vector p makes equal angles with the vectors a, b and c.

 

Posted by

infoexpert24

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