A line makes angles of 45∘ and 60∘ with the positive axes of X and Y respectively. The angle made by the same line with the positive axis of Z, is

The direction cosines of a line are given by the cosines of the angles made by the line with the positive directions of the x, y, and z axes.

Given that the line makes angles of 45° and 60° with the positive x and y axes respectively, we can find the direction cosines of the line as follows:

cos(45°) = √2/2 cos(60°) = 1/2

The sum of the squares of the direction cosines is equal to 1. Therefore, we can find the cosine of the angle made by the line with the positive z axis (cosZ) using the equation:

(cosX)^2 + (cosY)^2 + (cosZ)^2 = 1

Substituting the values we found for cosX and cosY:

(√2/2)^2 + (1/2)^2 + (cosZ)^2 = 1 1/2 + 1/4 + (cosZ)^2 = 1 (cosZ)^2 = 1 - 3/4 (cosZ)^2 = 1/4

Taking the square root of both sides, we find that cosZ = ±1/2.

Therefore, the angle made by the line with the positive z axis is either 60° or 120°.