Question

If non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are equally inclined to coplanar vector $\overrightarrow{c}$, then $\overrightarrow{c}$ can be

• A. $\frac{\left|\overrightarrow{a}\right|}{\left|\overrightarrow{a}\right|+2\left|\overrightarrow{b}\right|}\overrightarrow{a}+\frac{\left|\overrightarrow{b}\right|}{\left|\overrightarrow{a}\right|+\left|\overrightarrow{b}\right|}\overrightarrow{b}$
• B. $\frac{|\overrightarrow{b}|}{|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{a}+\frac{|\overrightarrow{a}|}{|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{b}$
• C. $\frac{|\overrightarrow{a}|}{|\overrightarrow{a}|+2|\overrightarrow{b}|}\overrightarrow{a}+\frac{|\overrightarrow{b}|}{|\overrightarrow{a}|+2|\overrightarrow{b}|}\overrightarrow{b}$
• D. $\frac{|\overrightarrow{b}|}{2|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{a}+\frac{\left|\overrightarrow{a}\right|}{2|\overrightarrow{a}|+\left|\overrightarrow{b}\right|}\overrightarrow{b}$

Answer 1

Answer: (B), (D)

$\frac{|\overrightarrow{b}|}{|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{a}+\frac{|\overrightarrow{a}|}{|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{b}$
$\frac{|\overrightarrow{b}|}{2|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{a}+\frac{\left|\overrightarrow{a}\right|}{2|\overrightarrow{a}|+\left|\overrightarrow{b}\right|}\overrightarrow{b}$

Since $\overrightarrow{a}$ and $\overrightarrow{b}$ are equally inclined to $\overrightarrow{c},\overrightarrow{c}$ must be of the form $t(\frac{\overrightarrow{a}}{|\overrightarrow{a}|}+\frac{\overrightarrow{b}}{|\overrightarrow{b}|})$
Now $\frac{|\overrightarrow{b}|}{|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{a}+\frac{|\overrightarrow{a}|}{|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{b}$
$=\frac{|\overrightarrow{a}||\overrightarrow{b}|}{|\overrightarrow{a}|+\overrightarrow{b}}(\frac{\overrightarrow{a}}{|\overrightarrow{a}|}+\frac{\overrightarrow{b}}{|\overrightarrow{a}|+|\overrightarrow{b}|})$
Also, $\frac{|\overrightarrow{b}|}{2|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{a}+\frac{|\overrightarrow{a}|}{2|\overrightarrow{a}|+|\overrightarrow{b}|}\overrightarrow{b}$
$=\frac{|\overrightarrow{a}|+|\overrightarrow{b}|}{2|\overrightarrow{a}|+\overrightarrow{b}}(\frac{\overrightarrow{a}}{|\overrightarrow{a}|}+\frac{\overrightarrow{b}}{|\overrightarrow{b}|})$
Other two vectors cannot be written in the form $t(\frac{\overrightarrow{a}}{|\overrightarrow{a}|}+\frac{\overrightarrow{b}}{|\overrightarrow{b}|})$