How would you find a unit vector parallel to the resultant of the vectors A = 2i - 6j - 3k and B = 4i + 3j - k?

To find a unit vector parallel to the resultant of two vectors, we can follow these steps:

Find the resultant vector by adding the two given vectors together: R = A + B.
Calculate the magnitude of the resultant vector, ||R||, using the formula: ||R|| = sqrt(Rx^2 + Ry^2 + Rz^2), where Rx, Ry, and Rz are the components of the resultant vector R.
Divide the resultant vector R by its magnitude ||R|| to obtain a unit vector parallel to R.
Let's apply these steps to the given vectors A = 2i - 6j - 3k and B = 4i + 3j - k:

Add the vectors A and B to find the resultant vector R:
R = A + B
= (2i - 6j - 3k) + (4i + 3j - k)
= 2i - 6j - 3k + 4i + 3j - k
= 6i - 3j - 4k
So, the resultant vector R is 6i - 3j - 4k.

Calculate the magnitude of the resultant vector ||R||:
||R|| = sqrt(Rx^2 + Ry^2 + Rz^2)
= sqrt((6)^2 + (-3)^2 + (-4)^2)
= sqrt(36 + 9 + 16)
= sqrt(61)

Divide the resultant vector R by its magnitude ||R|| to obtain a unit vector parallel to R:
Unit vector parallel to R = R / ||R||
= (6i - 3j - 4k) / sqrt(61)

So, the unit vector parallel to the resultant of vectors A and B is (6i - 3j - 4k) / sqrt(61).