Question

The sum of the magnitudes of two forces acting at a point is $18$ and the magnitude of their resultant is $12$. If the resultant is at ${90}^{\circ }$ with the force of smaller magnitude, then what are the magnitudes of the forces?

• A. $12,5$
• B. $14,4$
• C. $5,13$
• D. $10,8$

Answer 1

The correct option is C $5,13$

Let $A$ and $B$ be the two vectors.
As given in the problem,
$A+B=18$
$\therefore B=18-A-\left(1\right)$

Now, resultant is at ${90}^{\circ }$ to the smaller vector, $A$ (say)


From the diagram, $B\cos \theta =A$
and $B\sin \theta =R=12$

Squaring and adding, $B^2=A^2+R^2=A^2+{12}^2$
Using (1), $(18-A{)}^2=A^2+{12}^2$
$\Rightarrow {18}^2+A^2-36A=A^2+{12}^2$
$\Rightarrow 36A=180$
i.e $A=5$

$A+B=18$
$\Rightarrow B=13$
Thus, the magnitude of the two vectors are $5$ and $13$.