Do these points make a straight line?

Question

I'm trying to prepare for my calculus 3 class coming up this fall and doing some practice problems. I'm having a hard time visualizing some of these 3D coordinates.

$D(0,-5,5)$ $E(1,-2,4)$ $F(3,4,2)$

The answer is yes and I drew out the coordinates to try and understand why this answer is a yes and although I can't 'see' why it's a yes I imagine it is because all the points appear to be on the same plane, yz?

Is this correct and do you have any suggestions that can help visualizing or is it just a matter of practice?

Answer 1

If you're comfortable with the definition of vectors in $3\text D$, then you may create vectors $\vec{DE} = (1,3,-1)$ and $\vec{EF} = (2,6,-2)$. If $\vec{DE}$ is a scalar multiple of $\vec{EF}$, then visually, you'd have two vectors pointing in the same (or complete opposite) direction. Remember that geometrically, a vector is a straight arrow with magnitude and direction.

So find $c$ such that $$c(1,3,-1) = (2,6,-2).$$ The solution is then $c = 2$ because $$2(1,3,-1) = (2,6,-2).$$ This means that $\vec{DE}$ points in exactly the same direction as $\vec{EF}$ which means that they lie on the same line in space.

Answer 2

One way to check that all the points are co-linear is to verify that this determinant: $$\begin{vmatrix} 0 & - 5 & 5 \\ 1 & - 2 & 4 \\3 & 4 & 2\end{vmatrix} =0 .$$

Answer 3

To show the three points are on the same line, you can form vectors with their coordinates and check that the vectors are parallel. It is not enough to show they are on the same plane, because three points may be on the same plane and not be on the same line (in fact, any three non-colinear points determine a plane).

For example, the points (0,0,1), (0,1,0), and (1,0,0) are on the same plane, but you can see that they are not on the same line.

Now, in your example, you can form the following vectors:

$$DE=(1-0,-2-(-5),4-5)=(1,3,-1)\\ DF=(3-0,4-(-5),2-5)=(3,9,-3)$$

By definition,two vectors $u,v$ are parallel if you can find a real number $k$ such that $u=kv$.

Can you find a $k$ such that $(3,9,-3)=k(1,3,-1)$?