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), and (l3, m3, n3), we need to show that the dot product between the direction vector of the line and the direction vectors of the three given lines is equal.
Let's consider the direction vector (a, b, c) of the line whose direction ratios are l1 + l2 + l3, m1 + m2 + m3, and n1 + n2 + n3.
The dot product of this direction vector with the direction vector (l1, m1, n1) of the first line is given by:
(a * l1) + (b * m1) + (c * n1)
Similarly, the dot product of the direction vector with the direction vectors of the second and third lines can be calculated as:
(a * l2) + (b * m2) + (c * n2)
(a * l3) + (b * m3) + (c * n3)
Now, since the three given lines are mutually perpendicular, their direction vectors are orthogonal to each other.
Hence, the dot product of any two of these direction vectors will be zero.
Therefore, we can write the following equations:
(a * l1) + (b * m1) + (c * n1) = 0 ----(1)
(a * l2) + (b * m2) + (c * n2) = 0 ----(2)
(a * l3) + (b * m3) + (c * n3) = 0 ----(3)
Adding equations (1), (2), and (3), we have:
(a * (l1 + l2 + l3)) + (b * (m1 + m2 + m3)) + (c * (n1 + n2 + n3)) = 0
Therefore, the line whose direction ratios are l1 + l2 + l3, m1 + m2 + m3, and n1 + n2 + n3 makes equal angles with the lines having direction cosines (l1, m1, n1), (l2, m2, n2), and (l3, m3, n3).
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listen to me dear.solve these six equations.l1l2+m1m2+n1n2=0,l2l3+m2m3+n2n3=0and l1l3+m1m3+n1n3=0...with these solve theselil2+m1m2+n1n2=l2l3+m2m3+n2n3=l1l3+m1m3+n1n3
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Because the first three lines are mutually perpendicular, dot products of pairs of lines are zero.
l1*l2+m1*m2+n1*n2 = 0
l2*l3+m2*m3+n2*n3 = 0
l1*l3+m1*m3+n1*n3 = 0
Finally, using the above equations, and another dot product, show that the cosine of the angle between the fourth vector and any of the first three vectors is the same.