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.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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.

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