The vector v makes an angle of 30 degrees with the x-axis and equal angles with both the y and z axes. Determine the direction cosines for vector v and the angle that vector v makes with the z-axis.
Let the vector V = ai + bj + ck.
The vector makes an angle of 30 degrees with the x-axis and equal angles with the y and z axes.
If a vector V = ai + bj + ck makes an angle of A with the x-axis, B with the y=axis and C with the z-axis.
The direction cosines are cos A = `a/|p|` , cos B = `b/|p|` and cos C = `c/|p|` where `p = sqrt(a^2 + b^2 + c^2)`.
Also, `cos^2 A + cos^2 B + cos^2 C = 1`
As an equal angle is made with the y and z axes
=> `cos^2 A + 2*cos^2 B = 1`
=> `3/4 + 2*cos^2 B = 1`
=> `2*cos^2 B = 1/4`
=> `cos^2 B = 1/8`
=> `cos B = 1/(2*sqrt 2)`
The required direction cosines of the vector are cos A = `sqrt 3/2` where A is the angle made with the x axis and cos B = cos C = `1/(2*sqrt 2)` where B and C are the angles made with the y and z axes.