Question

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

• A. $27,-18,54$
• B. $-27,18,-54$
• C. $-27,-18,-54$
• D. $27,-18,-54$

Answer 1

The correct option is B $-27,18,-54$

Let the components of the line segment vector be $a,b,c$, then

$a^2+b^2+c^2={\left(63\right)}^2$ ...(1)

Also $\frac{a}{3}=\frac{b}{-2}=\frac{c}{2}=\lambda$(say)

Then $a=3\lambda ,b=-2\lambda$ and $c=6\lambda$ and from (1), we have

$9{\lambda }^2+4{\lambda }^2+36{\lambda }^2={\left(63\right)}^2$

$\Rightarrow 49{\lambda }^2={\left(63\right)}^2$

$\Rightarrow \lambda =\pm \frac{63}{7}=\pm 9$

Since $a=3\lambda <0$ as the line makes an obtuse angle with $x$-axis,

$\lambda =-9$ and the required components are $-27,18,-54$