You are here:

- Home
- Teens
- Homework/Study Tips
- Calculus
- Vectors and lines in 3-space

Advertisement

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.

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

Answers by Expert:

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.**Education/Credentials**

(See above.)