Alright, let's explore this concept step by step.
A unit vector is a vector of length 1 and in the same direction as the given vector. On the other hand, a collinear vector can be found along the same line or parallel lines.
Given an input vector (4, 1, -7), we first need to find the norm (or magnitude) of this vector. The norm is calculated as the square root of the sum of the squares of all the coordinates of the vector.
After calculating, we find the norm to be approximately 8.12404.
Next, we create a unit vector by dividing the original vector by its norm. This gives us the array [0.49236596, 0.12309149, -0.86164044].
The unit vector in the opposite direction, or the negative of the unit vector, is also collinear with the original vector. We can calculate this simply by multiplying the unit vector by -1. The result is the array [-0.49236596, -0.12309149, 0.86164044].
So, based on these calculations, the unit vectors collinear with (4, 1, -7) are approximately [0.49236596, 0.12309149, -0.86164044] and [-0.49236596, -0.12309149, 0.86164044].