Vector Basics

Finding Resultant Vectors

Vectors can be added together by drawing them, then the single vector that goes from the start of the first vector to the end of the final vector is known as the resultant vector.

We can see that to get to the end point of $boldsymbol{r}$, we need to go along the vector $boldsymbol{a}$ then the vector $boldsymbol{b}$.

Thus, $boldsymbol{r}=boldsymbol{a}+boldsymbol{b}$

The order that you add the vectors in doesn’t matter, the resultant vector is always the same.

Multiplying Vectors by Scalars

When you multiply a vector by a scalar (numeric value), this changes the length of the vector, but has no affect on the direction.

If a vector is multiplied by a non-zero scalar, the new vector is always parallel to the original vector, e.g. $boldsymbol{a}$ is parallel to $3boldsymbol{a}$

To show two vectors are parallel you need to show that they are scalar multiples of each other.

Example: $overrightarrow{AB}=boldsymbol{x}$, $overrightarrow{BC}=boldsymbol{y}$

The point $M$ lies on the line $AB$ and divides $overrightarrow{AB}$ in the ratio $3:2$ and $N$ lies on the line $BC$ and divides $overrightarrow{BC}$ in the ratio $2:3$

Show that $overrightarrow{MN}$ is parallel to $overrightarrow{AC}$

$overrightarrow{AC}=boldsymbol{x}+boldsymbol{y}$

$M$ divides $overrightarrow{AB}$ in the ratio $3:2$ so $M$ is $dfrac{3}{5}$ of the way along $overrightarrow{AB}$. Therefore, $overrightarrow{AM}=dfrac{3}{5}boldsymbol{x}$, so $overrightarrow{MB}=dfrac{2}{5}boldsymbol{x}$.

Similarly, $N$ is $dfrac{2}{5}$ of the way along $overrightarrow{BC}$, so $overrightarrow{BN}=dfrac{2}{5}boldsymbol{y}$

Finally, $overrightarrow{MN}=dfrac{2}{5}boldsymbol{x}+dfrac{2}{5}boldsymbol{y}=dfrac{2}{5}(boldsymbol{x}+boldsymbol{y})=dfrac{2}{5}overrightarrow{AC}$, which shows $overrightarrow{MN}$ is parallel to $overrightarrow{AC}$