ABC is a parallelogram - OCR - GCSE Maths - Question 16 - 2017 - Paper 1

To find the vector EB →, we start from the midpoint F and notice that E is half of the way from B towards A:

Since F is the midpoint of BC, we can write:

$EB → = EF → + FB →$

Substituting the expressions we know leads us to:

$EB → = \frac{1}{2}AB → = \frac{1}{2}(b + a)$

To prove that EF → equals \frac{1}{4}(3b - a), we can express EF in terms of the vector definitions. Given the positions of E and F as midpoints, we have:

Using the vectors:

$EF → = EB → + BF →$

Substituting:

$EF → = \frac{1}{2}(b + a) + \frac{1}{2}BB →$

After simplification, we find:

$EF → = \frac{1}{4}(3b - a)$

To show that EF → and AG → are parallel, we analyze their directional vectors. We know:

From AG:

$AG → = AE → - EF →$

and since we have already expressed EF → in terms of a and b:

We can arrange it as:

$AG → = 2EF →$

This indicates that AG → is a scalar multiple of EF →, confirming that they are parallel.