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Calculus/Vectors and lines in 3-space

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Question
If x,y and z are the angles that a line makes with x-axis, y-axis and z-axis respectively, then find the value of
cos 2x+cos 2y+cos 2z.

Answer
Questioner:Akshar
Country:Gujarat, India
Category:Calculus
Private:No
Subject:Mathematics

Question:
If x,y and z are the angles that a line makes with x-axis, y-axis and z-axis respectively, then find the value of
cos 2x+cos 2y+cos 2z.
.........................................................
The line has the equation:

P = P0 + t(V)

where:
P(x,y,z) is any point on the line.
P0 is some fixed point, which might as well be the origin.
t is a variable parameter.
V(a,b,c) is a direction vector.

I'll write your problem as:

If A,B,C  are the angles that a line makes with x-axis, y-axis and z-axis respectively, then find the value of
cos 2A + cos 2B + cos 2C.

The DIRECTION COSINES (look in the chapter on vectors) of the vector V are:

cos A = a/|V|,  cos B = b/|V|,  cos C = c/|V|,

and |V| = sqrt(a^2 + b^2 + c^2)

Now I think you know how to proceed.  From your calculus book: (Well, from mine, anyway)

Cos^2(A) = a^2/(a^2 + b^2 + c^2)
Cos^2(B) = b^2/(a^2 + b^2 + c^2)
Cos^2(C) = c^2/(a^2 + b^2 + c^2)

Cos^2(A) + Cos^2(B) + Cos^2(C) = 1  (YES!)

Now just use a trig formula for cos(2A,2B,2C) to get your answer.

cos 2A + cos 2B + cos 2C =

2 cos^2(A) - 1 + 2 cos^2(B) - 1 + 2 cos^2(C) - 1 =

2(1) - 3 = - 1

Cute problem.  It's probably someone's theorem.  

Calculus

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