if (l1,m1,n1),(l2,m2,n2),(l3,m3,n3) are the direction cosines of three mutual perpendicular lines,show that the line whose direction ratios area l1+l2+l3,m1+m2+m3,n1+n2+n3 make equal angles with them

To prove that the line whose direction ratios are (l1 + l2 + l3, m1 + m2 + m3, n1 + n2 + n3) makes equal angles with the lines having direction cosines (l1, m1, n1), (l2, m2, n2), (l3, m3, n3), we need to show that the dot product of the direction vector of the line we want to prove and the direction vectors of the given lines is equal to zero.

Let's consider the direction vector of the line we want to prove as A(l1 + l2 + l3, m1 + m2 + m3, n1 + n2 + n3). Now, let's calculate the dot product of A and the direction vectors of the given lines:

Dot product with the first line (l1, m1, n1):
A⋅(l1, m1, n1) = (l1 + l2 + l3)⋅l1 + (m1 + m2 + m3)⋅m1 + (n1 + n2 + n3)⋅n1
= l1^2 + m1^2 + n1^2 + l1⋅l2 + l1⋅l3 + m1⋅m2 + m1⋅m3 + n1⋅n2 + n1⋅n3

Dot product with the second line (l2, m2, n2):
A⋅(l2, m2, n2) = (l1 + l2 + l3)⋅l2 + (m1 + m2 + m3)⋅m2 + (n1 + n2 + n3)⋅n2
= l2^2 + m2^2 + n2^2 + l1⋅l2 + l2⋅l3 + m1⋅m2 + m2⋅m3 + n1⋅n2 + n2⋅n3

Dot product with the third line (l3, m3, n3):
A⋅(l3, m3, n3) = (l1 + l2 + l3)⋅l3 + (m1 + m2 + m3)⋅m3 + (n1 + n2 + n3)⋅n3
= l3^2 + m3^2 + n3^2 + l1⋅l3 + l2⋅l3 + m1⋅m3 + m2⋅m3 + n1⋅n3 + n2⋅n3

Now, let's add the above three dot products:

A⋅(l1, m1, n1) + A⋅(l2, m2, n2) + A⋅(l3, m3, n3)
= l1^2 + m1^2 + n1^2 + l2^2 + m2^2 + n2^2 + l3^2 + m3^2 + n3^2
+ l1⋅l2 + l1⋅l3 + l2⋅l3 + m1⋅m2 + m1⋅m3 + m2⋅m3 + n1⋅n2 + n1⋅n3 + n2⋅n3

We know that for three mutually perpendicular lines, their direction cosines satisfy the following properties:
l1⋅l2 + m1⋅m2 + n1⋅n2 = 0
l1⋅l3 + m1⋅m3 + n1⋅n3 = 0
l2⋅l3 + m2⋅m3 + n2⋅n3 = 0

Substituting these values in the above equation, we get:
A⋅(l1, m1, n1) + A⋅(l2, m2, n2) + A⋅(l3, m3, n3) = l1^2 + m1^2 + n1^2 + l2^2 + m2^2 + n2^2 + l3^2 + m3^2 + n3^2 = 0

Since the sum of the dot products of the line we want to prove with the direction vectors of the given lines is zero, it means that the line makes equal angles with each of the lines having direction cosines (l1, m1, n1), (l2, m2, n2), (l3, m3, n3).

Hence, it has been proved that the line with direction ratios (l1 + l2 + l3, m1 + m2 + m3, n1 + n2 + n3) makes equal angles with the lines having direction cosines (l1, m1, n1), (l2, m2, n2), (l3, m3, n3).