Maths with David

    9709. P3. Vectors. Past Exam Questions

    9709/31/M/J/25q8 – Mark Scheme

    With respect to the origin O, the points A and B have position vectors 2i + 4k and 5i + j + 6k respectively. The line l1 passes through the points A and B.

    (a) Find a vector equation for line l1 . [2 marks]

    The line l2 has equation r = 2i + j + 5k = \mu (i + 2j + 3k).

    (b) Show that l1 and l2 do not intersect. [4 marks]

    (c) Find the acute angle between the directions of l1 and l2

    9709/32/M/J/25q9 – Mark Scheme

    With respect to the origin O, the points A, B and C have position vectors given by \overrightarrow{OA} = \begin{pmatrix} 1 \\ -4 \\ 2 \end{pmatrix} , \overrightarrow{OB} = \begin{pmatrix} -1 \\ 1 \\ 3 \end{pmatrix} and \overrightarrow{OC} = \begin{pmatrix} 2 \\ 3 \\ 5 \end{pmatrix} .

    (a) Find a vector equation for the line through A and B. [2 marks]

    (b) Using a scalar product, find the exact value of cosBAC. [4 marks]

    (c) Hence find the exact area of triangle ABC. [3 marks]

    9709/33/M/J/25q9 – Mark Scheme

    With respect to the origin O, the points A, B and C have position vectors given by \overrightarrow{OA} = \textbf{i} + 2 \textbf{j} , \overrightarrow{OB} = \textbf{i} + 3 \textbf{j} - 2 \textbf{k} and \overrightarrow{OC} = 2 \textbf{i} - \textbf{j} + 3 \textbf{k} .

    The line l passes through B and C.

    (a) Find a vector equation for l. [2 marks]

    (b) The point P is the foot of the perpendicular from A to l.

    Find the position vector of P. [4 marks]

    (c) The point D is the reflection of A in l.

    Find the position vector of D. [2 marks]