2021 SAJC P2 Q3
2021 SAJC P2 Q3
10 marks
Prelims
The line \(L\) has equation \(\mathbf{r}=-\mathbf{i}+2\mathbf{j}-4\mathbf{k}+\lambda \left( 6\mathbf{i}+3\mathbf{j}-2\mathbf{k} \right)\), where \(\lambda \in \mathbb{R}\).
- Find the acute angle between \(L\) and the \(x\)-axis.[2]
The point \(P\) has position vector \(6\mathbf{i}+3\mathbf{j}-5\mathbf{k}\).
- Find the points on \(L\) which are at a distance of \(\sqrt{59}\) units from \(P\).Hence or otherwise find the point on \(L\) which is closest to \(P\).[5]
- Find a cartesian equation of the plane that includes the line \(L\) and the point \(P\).[3]
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Answer:(i) \({{31.0}^{\circ }}\) (ii) The points are \(\left( 11,\,\,8,\,\,-8 \right)\) or \(\left( -\frac{73}{49},\frac{86}{49},-\frac{188}{49} \right)\). Hence, the point closest to \(P\) is \(\left( \frac{233}{49},\frac{239}{49},-\frac{290}{49} \right)\)
(iii) \(x+8y+15z=-45\)
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