October 2020 Paper 3 Q4

OCR MEICurrent spec3 marksVectors

4 Fig. 4 shows the regular octagon ABCDEFGH.

Fig. 4: regular octagon ABCDEFGH with AB horizontal at the bottom, CD vertical on the right, EF horizontal at the top and GH vertical on the left — Fig. 4

\(\overrightarrow{\mathrm{AB}} = \mathbf{i}\), \(\overrightarrow{\mathrm{CD}} = \mathbf{j}\), where \(\mathbf{i}\) is a unit vector parallel to the \(x\)-axis and \(\mathbf{j}\) is a unit vector parallel to the \(y\)-axis.

Find an exact expression for \(\overrightarrow{\mathrm{BC}}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [3]

Mark scheme

Scheme	Marks	AO
BC is parallel to \(\mathbf{i} + \mathbf{j}\) OR 1 unit at 45°	M1	3.1a
\(\|\mathbf{i} + \mathbf{j}\| = \sqrt{2}\)	M1	1.1
\(\overrightarrow{\mathrm{BC}} = \dfrac{1}{\sqrt{2}}(\mathbf{i} + \mathbf{j})\) oe	A1	2.2a
	[3]

Notes

M1: Eg \(k\mathbf{i} + k\mathbf{j}\)

M1: Could be on diagram

A1: Must be exact eg \(\cos 45\,\mathbf{i} + \sin 45\,\mathbf{j}\)