Question

Let the coordinates of a points $p$ with respect to the system of non-coplanar vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ is $(3,2,1)$. Then coordinates of $p$ with respect to the system of vectors $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c},\overrightarrow{a}-\overrightarrow{b}+\overrightarrow{c}$ and $\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}$ is

• A. $(\frac{3}{2},\frac{1}{2},1)$
• B. $(\frac{3}{2},1,\frac{1}{2})$
• C. $(\frac{1}{2},\frac{3}{2},1)$
• D. $(1,\frac{3}{2},\frac{1}{2})$

Answer 1

The correct option is A $(\frac{3}{2},\frac{1}{2},1)$

coordinate of p with respect to $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}=(3,2,1)$

$\therefore (3,2,1)=3\overrightarrow{a}+2\overrightarrow{b}+\overrightarrow{c}$ ___ (1)

Now, $\alpha (\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})+\beta (\overrightarrow{a}-\overrightarrow{b}+\overrightarrow{c})+\gamma (\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c})$

$=\alpha \overrightarrow{a}+\alpha \overrightarrow{b}+\alpha \overrightarrow{c}+\beta \overrightarrow{a}-\beta \overrightarrow{b}+\beta \overrightarrow{c}+\gamma \overrightarrow{a}+\gamma \overrightarrow{b}-\overrightarrow{c}$

$=(\alpha +\beta +\gamma )\overrightarrow{a}+(\alpha -\beta +\gamma )\overrightarrow{b}+(\alpha +\beta -\gamma )\overrightarrow{c}$ __ (ii)

comparing (i) and (ii), we get

$\alpha +\beta +\gamma =3$ ___ (iii)

$\alpha -\beta +\gamma =2$ ___ (iv)

$\alpha +\beta -\gamma =1$ ___ (v)

$\left(iii\right)-\left(iv\right)\Rightarrow 2\beta =1$

$\Rightarrow \beta =\frac{1}{2}$

$\left(iv\right)+\left(v\right)\Rightarrow 2\alpha =3$

$\alpha =\frac{3}{2}$

$\left(iii\right)+\left(iv\right)\Rightarrow 2\alpha +2\gamma =5$

$2\left(\frac{3}{2}\right)+2\gamma =5$

$3+2\gamma =5$

$\Rightarrow 2\gamma =5-3$

$\Rightarrow 2\gamma =2$

$\Rightarrow \gamma =1$

$\therefore$ coordinate of p with respect to $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c},\overrightarrow{a}-\overrightarrow{b}+\overrightarrow{c}$ and

$\overrightarrow{a}+\overrightarrow{b}-\overrightarrow{c}=(\alpha ,\beta ,\gamma )$

$=(\frac{3}{2},\frac{1}{2},1)$