Vector worked examples

Example 1

Write in terms of $a$ and $b$ the vector $vec (BC)$

Write in terms of a and b with the following vectors

Answer as follows:

Think of this as two separate vectors

Separate the vectors in 2

Therefore

$vec (BC)=-a+b$

Connect up the 2 vectors and by reversing the direction of a to connect up to B

Or $vec (BC)=b-a$

Join B with a by its end point

Example 2

Find the line B and d with vectors M and N

$vec (AB)=m$

$vec (AD)=n$

The point $C$ on $BD$ is such that:

$BC:CD=1:3$

i. Find $D$ to $B$ in terms of $n$ and $m$

Think of this as the separate vectors

Separate the vectors in 2

Therefore

$vec (DB)=-n+m$

Reverse N so it can connect up with M

Or $vec (DB)=m-n$

Flip the 2 vectors so n is above m

$vec (DB)=-n+m=m-n$ (The latter is chosen because it's neater)

Remember don’t forget the arrow

ii. Find $vec (AC)$ in terms of $m$ and $n$

remember

Remember the line BCD

We are told $BC:CD=1:3$

this means

Measurements of BC and CD is 1 and 3

which also means

BC is a quarter of BD and CD is 3 quarters of BD

which also means

Which also means BC is a quarter n and CD is 3 quarters n

So if we redraw the original diagram

Redraw the shape showing the new vector

$vec (AC)$ in terms of $m$ and $n$ are:

$vec (AC)=n+3/4m-3/4n$

$vec (AC)=3/4m+1/4n$

OR

$vec (AC)=m-(1/4m-1/4n)$

$vec (AC)=m+(-1times(1/4m-1/4n))$

$vec (AC)=m-1/4m+1/4n$

$vec (AC)=3/4m+1/4n$

Both get the same answer (as they should)

Answer:

$vec (AC)$ in terms of $m$ and $n=3/4m+1/4n$

Example 3

Now try this example using the parallel lines to work it out

If $AD$ is parallel to $CB$ and $AB$ is parallel to $DC$ , write in terms of $a$ and $b$ the vectors $vec (BC)$ and $vec (BA)$

Answer as follows:

Think of this as separate vectors

Separate the vectors

Therefore

$vec (BC)=a$

and

$vec (BA)=-c$