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WAEC 2008 · Paper 2 · Q7

The coordinates of points XX, YY and ZZ are (4,0)(4, 0), (6,2)(6, 2) and (−2,1)(-2, 1) respectively. Find:

  1. (a)

    2XY→+3YZ→2\overrightarrow{XY} + 3\overrightarrow{YZ};

    Separate values with commas, e.g. 3, −2

  2. (b)

    the unit vector in the direction of XZ→\overrightarrow{XZ}.

    Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

  1. XY→=Y−X\overrightarrow{XY} = Y - X
    =(6−42−0)= \begin{pmatrix} 6 - 4 \\ 2 - 0 \end{pmatrix}
    =(22)= \begin{pmatrix} 2 \\ 2 \end{pmatrix}.
  2. YZ→=Z−Y\overrightarrow{YZ} = Z - Y
    =(−2−61−2)= \begin{pmatrix} -2 - 6 \\ 1 - 2 \end{pmatrix}
    =(−8−1)= \begin{pmatrix} -8 \\ -1 \end{pmatrix}.
  3. 2XY→+3YZ→=(44)+(−24−3)2\overrightarrow{XY} + 3\overrightarrow{YZ} = \begin{pmatrix} 4 \\ 4 \end{pmatrix} + \begin{pmatrix} -24 \\ -3 \end{pmatrix}
    =(−201)= \begin{pmatrix} -20 \\ 1 \end{pmatrix}.

(b)

  1. XZ→=Z−X\overrightarrow{XZ} = Z - X
    =(−61)= \begin{pmatrix} -6 \\ 1 \end{pmatrix}.
  2. Its magnitude is ∣XZ→∣=36+1|\overrightarrow{XZ}| = \sqrt{36 + 1}
    =37= \sqrt{37}.
  3. Divide the vector by its magnitude: the unit vector is 137(−6i+j)\dfrac{1}{\sqrt{37}}(-6\mathbf i + \mathbf j).
  4. Rationalise: 3737(−6i+j)\dfrac{\sqrt{37}}{37}(-6\mathbf i + \mathbf j).

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