Question

The vectors $a,b$ and $c$ are of the same length and, taken pairwise, they form equal angles. If $a=i+j$ and $b=j+k,$ the coordinates of $c$ are

• A. $(1,0,1)$
• B. $(1,2,3)$
• C. $(-1,1,2)$
• D. $(-1/3,4/3,1/3).$

Answer 1

Answer: (A), (D)

$(1,0,1)$
$(-1/3,4/3,1/3).$

Let $c=(c_1,c_2,c_3)$ Then $\left|c\right|=\left|a\right|=\left|b\right|=\sqrt{2}=\sqrt{c_1^2+c_2^2+c_3^2}$ It is given that the angles between the vectors are identical, and equal to $\varphi$ (say). Then
$\cos \varphi =\frac{a\cdot b}{\left|a\right|\left|b\right|}=\frac{0+1+0}{\sqrt{2}\sqrt{2}}=\frac{1}{2}$
$\frac{a\cdot c}{\left|a\right|\left|c\right|}=\frac{c_1+c_2}{\sqrt{2}\sqrt{2}}=\frac{1}{2}$
and $\frac{b\cdot c}{\left|b\right|\left|c\right|}=\frac{c_2+c_3}{2}=\frac{1}{2}$
Hence $c_1+c_2=1$ and $c_2+c_3=1$ That is $c_1=1-c_2$ and
$c_3=1-c_2$
$\Rightarrow 2=c_1^2+c_2^2+c_3^2\Rightarrow 2=(1-c_2{)}^2+c_2^2+(1-c_2{)}^2$
$\Rightarrow 2=3c_2^2+2-4c_2$
Therefore, $c_2=0$or $c_2=4/3.$ If $c_2=0$, then $c_1=1$ and $c_3=1$
and if $c_2=4/3,$ then $c_1=-1/3,c_3-1/3.$ Hence the coordinates of $c$ are
$(1,0,1)$ or $(-1/3,4/3,-1/3)$