andwith X-axis and Y-axis.4.a line makes angles α, β, γ with coordi

1.	andwith X-axis and Y-axis.4.a line makes angles α, β, γ with coordixes, prove that
Answer» we know that sum of squares of direction cosines of a vector is 1 so cos²A + cos²B +cos²C = 1 => 1-sin²A + 1-sin²B + 1 - sin²C = 1=> 3 - (sin²A +sin²B+sin²C ) = 1=> (sin²A +sin²B+sin²C ) = 3-1 = 2 and cos2A = 1-2sin²A so, cos2A +cos2B +cos2C = 1-2sin²A +1-2sin²B +1-2sin²C = 3-2(sin²A +sin²B+sin²C ) = 3-2(2) = -1.

andwith X-axis and Y-axis.4.a line makes angles α, β, γ with coordixes, prove that

Answer»

we know that sum of squares of direction cosines of a vector is 1

so cos²A + cos²B +cos²C = 1

=> 1-sin²A + 1-sin²B + 1 - sin²C = 1=> 3 - (sin²A +sin²B+sin²C ) = 1=> (sin²A +sin²B+sin²C ) = 3-1 = 2

and cos2A = 1-2sin²A

so, cos2A +cos2B +cos2C = 1-2sin²A +1-2sin²B +1-2sin²C = 3-2(sin²A +sin²B+sin²C ) = 3-2(2) = -1.