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  4. A line segment has length 63 and direction ratios, <3, -2, 6>. If the line makes an obtuse angle with x-axis, then the components of the line vector are

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Q. A line segment has length 63 and direction ratios, <3, -2, 6>. If the line makes an obtuse angle with x-axis, then the components of the line vector are

Introduction to Three Dimensional Geometry

Solution:

Let the components of the line segment vector be $a, b, c,$ then
$ a^2 + b^2 + c^2 = (63)^2$ ...(1)
Also $\frac{a}{3} = \frac{b}{-2} = \frac{c}{6} = \lambda$ (say) then $a = 3\lambda, b = - 2 \lambda $ and $c = 6 \lambda$. and from (1) , we have
$9\lambda^2 + 4\lambda^2 + 36 \lambda^2 = (63)^2 \, \Rightarrow \, 49 \lambda^2 = (63)^2 $
$\lambda = \pm \frac{63}{7} = \pm 9$.
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.