Let a = i + j + k , b = i and c =c_1 i +c_2 j +c_3 k . Then, If C… - Vidyadip

Question

Let $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}$ and $\vec{\text{c}}=\text{c}_1\hat{\text{i}}+\text{c}_2\hat{\text{j}}+\text{c}_3\hat{\text{k}}.$ Then,
If $C_2 = -1$ and $C_3 = 1$, show that no value of $C_1$ can make $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ coplanar.

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Answer

If $C_2= -1$ and $C_3 = 1$, then $\vec{\text{a}}=\hat{\text{i}}+\vec{\text{j}}+\vec{\text{k}},\vec{\text{b}}=\hat{\text{i}}$ and $\vec{\text{c}}={\text{c}}_1\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}.$
We know that vectors $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are coplanar if $\big[\vec{\text{a}}\vec{\text{b}}{\text{c}}\big]=0.$
for $\big[\vec{\text{a}},\vec{\text{b}},\text{c}\big]$ to be coplanar:
$\therefore\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]=0$
$\Rightarrow\begin{vmatrix}1&1&1\\1&0&0\\\text{c}_1&-1&1 \end{vmatrix}=0$
$\Rightarrow 1(0-0)-1(1-0)+1(-1-0)=0$
$\Rightarrow-1-1=0$
$\Rightarrow -2=0$
But this is never possible, whatever be the value of $C_1$. Thus, no value of $C_1$ can make $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ coplanar.

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