Geometry

YOU ARE LEARNING:

Vector Geometry

We can understand vectors as geometrical objects, and therefore deduce geometrical properties about them.

It is important to have a geometrical understanding of vectors. In this lesson you will learn about the resultant of a vector as well as how to calculate the midpoint of a vector.

We can also calculate the midpoint of a vector. To do this, we can multiply the vector by $\dfrac{1}{2}$ .

We can use vectors to prove geometrical facts, such as proving that three points are collinear.

For example, the vector $AB = \binom{3}{9}$ and $BC = \binom{4}{ 12}$ . Prove that these vectors are collinear.

First, show that they are parallel

Resultant vectors are parallel if one is a multiple of the other. This is because the individual $x$ and $y$ vectors remain in the same proportion.

BC is one third bigger than AB. Therefore, what do you have to multiply AB by to get to BC?

BC is $\dfrac{4}{3} \times AB$

Since BC is a multiple of AB, the vectors are definitely parallel.

Now we need to show the vectors share a point

Once we know this, we can conclude that the points are collinear, meaning that they are located on the same line.

Which letter do the two vectors share?

Both vectors pass through point B

Since they share a point, and are parallel, we can conclude that the two vectors are collinear.