Evaluating Vector Sums and Differences Numerically
Evaluating Vector Sums and Differences Numerically
Secondary 4
In the diagram, \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) represent vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively. \(X\), \(Y\) and \(Z\) are points such that \(\overrightarrow{OX} = \frac{3}{2}\overrightarrow{OB}\), \(\overrightarrow{AY} = \frac{2}{3}\overrightarrow{AB}\) and \(\overrightarrow{OZ} = \lambda\overrightarrow{OA}\).
- Express \(\overrightarrow{OX}\) and \(\overrightarrow{OY}\) in terms of \(\mathbf{a}\) and/or \(\mathbf{b}\).
- Express \(\overrightarrow{XZ}\) in terms of \(\lambda\), \(\mathbf{a}\) and \(\mathbf{b}\).
- Given that \(\lambda = \frac{3}{5}\), find the ratio of \(XY : XZ\).
- Find \(\frac{\text{area of}\hspace{0.5em} \Delta OYZ}{\text{area of}\hspace{0.5em} \Delta AYZ}\).
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Answer:(a) \(\overrightarrow{OX}=\dfrac32\mathbf b,\ \overrightarrow{OY}=\dfrac13\mathbf a+\dfrac23\mathbf b\) (b) \(\overrightarrow{XZ}=\lambda\mathbf a-\dfrac32\mathbf b\) (c) \(5:9\) (d) \(\dfrac32\)
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