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A line segment has length 63 and direction ratios are \(3,-2,6\). If the line makes an obtuse angle with \(\mathrm{x}\)-axis, the components of the line vector are
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\(-27,18,-54\)
Let the components of the line vector be a, b, c. Then \(\mathrm{a}^2+\mathrm{b}^2+\mathrm{c}^2=(63)^2\) ...(i)
Also \(\frac{\mathrm{a}}{3}=\frac{\mathrm{b}}{-2}=\frac{\mathrm{c}}{6}=\lambda\) (say), then \(\mathrm{a}=3 \lambda\), \(\mathrm{b}=-2 \lambda\) and \(\mathrm{c}=6 \lambda\) and from (i) we have \(9 \lambda^2+4 \lambda^2+36 \lambda^2=(63)^2\)
\(\Rightarrow 49 \lambda^2=(63)^2\)
\(\Rightarrow \lambda= \pm \frac{63}{7}= \pm 9\)
Since \(a=3 \lambda < 0\) as the line makes an obtuse angle with \(x\)-axis, \(\lambda=-9\) and the required components are \(-27,18,-54\).
Also \(\frac{\mathrm{a}}{3}=\frac{\mathrm{b}}{-2}=\frac{\mathrm{c}}{6}=\lambda\) (say), then \(\mathrm{a}=3 \lambda\), \(\mathrm{b}=-2 \lambda\) and \(\mathrm{c}=6 \lambda\) and from (i) we have \(9 \lambda^2+4 \lambda^2+36 \lambda^2=(63)^2\)
\(\Rightarrow 49 \lambda^2=(63)^2\)
\(\Rightarrow \lambda= \pm \frac{63}{7}= \pm 9\)
Since \(a=3 \lambda < 0\) as the line makes an obtuse angle with \(x\)-axis, \(\lambda=-9\) and the required components are \(-27,18,-54\).
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