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# Vector Geometry

### 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**.