# Vector worked examples

**Example 1**

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

**Answer as follows:**

Think of this as two separate vectors

Therefore

`vec (BC)=-a+b`

Or `vec (BC)=b-a`

**Example 2**

`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

Therefore

`vec (DB)=-n+m`

Or `vec (DB)=m-n`

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

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

remember

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

this means

which also means

which also means

So if we redraw the original diagram

`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**

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

Therefore

`vec (BC)=a`

and

`vec (BA)=-c`