Question

Calculate the scalar product of the following vectors.
Resolve the vector $d=\{1,1,1\}$ into components with respect to three noncoplanar vectors $a=\{1,1,-2\},b=\{1,-1,0\},andc=\{0,2,3\}.$

Answer 1

Given $\overrightarrow{a}=(1,1,-2),\overrightarrow{b}=(1,-,1,0),\overrightarrow{c}=(0,2,3),\overrightarrow{d}=(1,1,1)$

Let $d=x\overrightarrow{a}+y\overrightarrow{b}+z\overrightarrow{c}$

$(1,1,1)=x(1,1,-2)+y(1,-1,0)+z(0,2,3)$

Comparing respective components,we get

$x+y=\mathrm{1.............}\left(1\right)$

$x-y+2z=\mathrm{1............}\left(2\right)$

$-2x+3z=\mathrm{1..........}\left(3\right)$

Adding $\left(1\right)+\left(2\right)+\left(3\right)$

we get $z=\frac{3}{5}$

substituting z in $\left(2\right)$, we get

$x-y=-\frac{1}{5}.........\left(4\right)$

By adding $\left(1\right)+\left(4\right)$ ,we get

$x=\frac{2}{5}$

And by substituting x in (1), we get $y=\frac{3}{5}$

$\overrightarrow{d}=\frac{2}{5}\overrightarrow{a}+\frac{3}{5}\overrightarrow{b}+\frac{3}{5}\overrightarrow{c}$