Find the sum of each of the component vectors to determine the resultant vector. Use the following notation to express vectors: Ai + Bj + Ck, where i, j and k are units vectors pointing in the direction of the positive x, y and z axes respectively. A, B and C are the magnitudes in each of those directions. Adding vectors is simply a matter of finding the sum of each of the coefficients. For example: (2i + 2j + 2k) + (2i + 3j + 4k) = 4i + 5j + 6k.
Calculate the magnitude of the resultant vector using the Pythagorean theorem. This theorem states that the length of a diagonal is the square root of the sum of the squares of the sides. You can picture the coefficients of a vector as the lengths of a box's sides, and the resultant vector is a diagonal extending across opposite corners of the box. Square each of the coefficients, add them up and find the square root. For example, the magnitude of the vector 4i + 5j + 6k is (4^2 + 5^2 + 6^2)^1/2 = 8.77.
Find the direction cosines with respect to each of the axes. The cosine of the angle that the vector forms with respect to a given axis equals the magnitude of the component vector along that axis divided by the overall magnitude. Expressing that for the x-axis: cos(Ax) = Mx/M, where Ax is the angle with respect to the x-axis, Mx is the component magnitude along the x-axis and M is the overall magnitude. For example, the magnitude of the vector 4i + 5j + 6k along the y-axis is 5, so the cosine of the angle that the vector makes with the y-axis is cos(Ay) = 5/8.77 = 0.570. Therefore the angle with respect to the y-axis is arccos(0.570) = 55.2 degrees.