Question

Don't really know how to approach this, a little guidance would be appreciated.

Answer 1

Determine whether the points lie on a straight line:

A(2, 4, 2), B(3, 7, -2), C(1, 3, 3)

Answer 2

Hmm, lets see. First stretch two vectors from, frist A to B, second A to C. Like this:
If these are equal, they are on the same line:
\[AB . AC = \left| AB \right| \times \left| AC \right| \times cos \theta\]

\[\cos \theta = \frac{ AB.AC }{ \left| AB \right|.\left| AC \right| }\]
The dot product of this two vectors will tell us if they are on the same line. First define the vectors AB and AC, and calculate their dot product.

If cosine of the angle theta equals to 1, they are on the same line.

Answer 3

By the way, this notation:\[AB\]
shows a vector from A to B. I couldn't find the vector sign, which is an arrow on top of them.

Answer 4

That makes sense, but is there another way to approach it? I know Dot product, but that is a few lessons ahead of what we're suppose to know.

Answer 5

If this is was a two dimensional space, maybe we can approach this by first find the equation of a line going through A and B or A and C. And check if the third point is in the line. The same method can also be applied here but in order to find the equation of the line here, you will need dot product again. In addition, the line equation in 3 dimensional space is parametric. Like this:
x = 3t +3
y = 2t
z = t
... for example.

Answer 6

Dot product is easy to learn. It is the very basic of vector operations.
https://www.mathsisfun.com/algebra/vectors-dot-product.html