4. Relative to a fixed origin, two lines have the equations
$$\begin{aligned}
& \mathbf { r } = \left( \begin{array} { c }
7 \\
0 \\
- 3
\end{array} \right) + \lambda \left( \begin{array} { c }
5 \\
4 \\
- 2
\end{array} \right)
\end{aligned}$$
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(a) \(4\vec{d} = 6 + 14\mu\) (1)B1
\(-3 - 2\lambda = 3 + 2\mu\) (2) \((1) + 2 \times (2): -6 = 12 + 18\mu, \mu = -1, \lambda = -2\) M1 A1
\(\vec{r} = \begin{pmatrix} 7 \\ 0 \\ -3 \end{pmatrix} - 2\begin{pmatrix} 5 \\ -2 \\ 1 \end{pmatrix} = \begin{pmatrix} -3 \\ -8 \\ -1 \end{pmatrix}\) M1 A1
(b) \(a - (-5) = -3, a = -8\)M1 A1
(c) \(\cos \theta = \frac{5×(-5) + 4×[4 + (-2)×2]}{\sqrt{25+16+4×\sqrt{25+196+4}}}\)M1 A1
\(= \frac{-27}{\sqrt{45×15}} = \frac{9}{3\sqrt{5}×5} = \frac{3}{5\sqrt{5}} = \frac{3}{25}\sqrt{5}\) M1 A1
(11)
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**(a)** $4\vec{d} = 6 + 14\mu$ (1) | B1 |
$-3 - 2\lambda = 3 + 2\mu$ (2) | |
$(1) + 2 \times (2): -6 = 12 + 18\mu, \mu = -1, \lambda = -2$ | M1 A1 |
$\vec{r} = \begin{pmatrix} 7 \\ 0 \\ -3 \end{pmatrix} - 2\begin{pmatrix} 5 \\ -2 \\ 1 \end{pmatrix} = \begin{pmatrix} -3 \\ -8 \\ -1 \end{pmatrix}$ | M1 A1 |
**(b)** $a - (-5) = -3, a = -8$ | M1 A1 |
**(c)** $\cos \theta = \frac{5×(-5) + 4×[4 + (-2)×2]}{\sqrt{25+16+4×\sqrt{25+196+4}}}$ | M1 A1 |
$= \frac{-27}{\sqrt{45×15}} = \frac{9}{3\sqrt{5}×5} = \frac{3}{5\sqrt{5}} = \frac{3}{25}\sqrt{5}$ | M1 A1 | (11)
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4. Relative to a fixed origin, two lines have the equations
$$\begin{aligned}
& \mathbf { r } = \left( \begin{array} { c }
7 \\
0 \\
- 3
\end{array} \right) + \lambda \left( \begin{array} { c }
5 \\
4 \\
- 2
\end{array} \right)
\end{aligned}$$
\hfill \mbox{\textit{Edexcel C4 Q4 [11]}}