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If $a=2\overset{\wedge }{\mathop{i}}\,-\overset{\wedge }{\mathop{j}}\,-m\overset{\wedge }{\mathop{k}}\,\text{ and }b=\dfrac{4}{7}\overset{\wedge }{\mathop{i}}\,-\dfrac{2}{7}\overset{\wedge }{\mathop{j}}\,+2\overset{\wedge }{\mathop{k}}\,$ are collinear, then the value of m is equal to
(A) -7
(B) -1
(C) 2
(D) 7
(E) -2

Last updated date: 17th Jul 2024
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Hint: Two vectors are collinear if they are in the same line. Use formula $\overset{\to }{\mathop{A}}\,.\overset{\to }{\mathop{B}}\,=\left( \overset{\to }{\mathop{A}}\, \right)\left( \overset{\to }{\mathop{B}}\, \right)\cos \theta $ where $\theta $is angle between $\overset{\to }{\mathop{A}}\,\text{ and }\overset{\to }{\mathop{B}}\,$.

Complete step-by-step answer:
Here, we have two vectors given as $\overset{\to }{\mathop{a}}\,=2i-j-mk\text{ and }\overset{\to }{\mathop{b}}\,=\dfrac{4}{7}i-\dfrac{2}{7}j+2k.$
Now, we have to find value of m if $\overset{\to }{\mathop{a}}\,\text{ and }\overset{\to }{\mathop{b}}\,$ are collinear.
As, we know collinear means in a line, that is if two vectors $\overset{\to }{\mathop{a}}\,\text{ and }\overset{\to }{\mathop{b}}\,$ are collinear then they lie in the same line.
In another language, we can say that if two vectors are collinear then the angle between them is $0{}^\circ $.
Hence, $\overset{\to }{\mathop{a}}\,\text{ and }\overset{\to }{\mathop{b}}\,$ have angle $0{}^\circ $ between them.
Now, we know dot product of two vectors can be given by;
$\overset{\to }{\mathop{A}}\,.\overset{\to }{\mathop{B}}\,=\left( \overset{\to }{\mathop{A}}\, \right)\left( \overset{\to }{\mathop{B}}\, \right)\cos \theta $
Where $\theta $ is angle between $\overset{\to }{\mathop{A}}\,\text{ and }\overset{\to }{\mathop{B}}\,$.
As, we have two vectors $\overset{\to }{\mathop{a}}\,=2i-j-mk\text{ and }\overset{\to }{\mathop{b}}\,=\dfrac{4}{7}i-\dfrac{2}{7}j+2\overset{\wedge }{\mathop{k}}\,$ which have $0{}^\circ $ between them.
$\overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{b}}\,=\left| \overset{\to }{\mathop{a}}\, \right|\left| \overset{\to }{\mathop{b}}\, \right|\cos \theta $………………….(1)
We know magnitude of any vector $\overset{\to }{\mathop{r}}\,=xi-yj-zk$ is given as;
$\left| \overset{\to }{\mathop{r}}\, \right|=\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}$
Hence, equation (1) can be written as
We know that $\cos 0{}^\circ =1$ and can simplify the above equation as;
  & \overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{b}}\,=\sqrt{5+{{m}^{2}}}\sqrt{\dfrac{20}{49}+4}\left( 1 \right) \\
 & \overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{b}}\,=\sqrt{5+{{m}^{2}}}\dfrac{\sqrt{216}}{7} \\
 & or \\
 & \overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{b}}\,=\dfrac{6\sqrt{6}}{7}\sqrt{5+{{m}^{2}}}.............\left( 2 \right) \\
Now, we can rewrite $\overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{b}}\,$ in different way as;
$\overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{b}}\,=\left( 2i-j-mk \right).\left( \dfrac{4}{7}i-\dfrac{2}{7}j+2k \right)............\left( 3 \right)$
As we know the same vectors have angle $0{}^\circ $. Hence, angles between (i, j), (j, k) and (i, k) are perpendicular to each other as they are unit vectors through the x, y and z axis.
Hence, equation (3) can be solved as;
  & \overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{b}}\,=2\times \dfrac{4}{7}+1\times \dfrac{2}{7}-m\times 2 \\
 & \overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{b}}\,=\dfrac{10}{7}-2m..............\left( 4 \right) \\
Now, equation (2) and (4) have equal L.H.S hence R.H.S should also be equal to each other.
$\overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{b}}\,=\dfrac{6\sqrt{6}}{7}\sqrt{5+{{m}^{2}}}=\dfrac{10}{7}-2m$
Squaring both sides of the equation we get;
  & \dfrac{216\left( 5+{{m}^{2}} \right)}{49}=\dfrac{{{\left( 10-14m \right)}^{2}}}{49} \\
 & 216\times 5+216{{m}^{2}}=100+196{{m}^{2}}-280m \\
 & 20{{m}^{2}}+280m+980=0 \\
Dividing the whole equation by 20, we get;
Now, we can factorize above equation as;
${{\left( m+7 \right)}^{2}}=0$
  & m+7=0 \\
 & m=-7 \\
Therefore, for m = -7, $\overset{\to }{\mathop{a}}\,\text{ and }\overset{\to }{\mathop{b}}\,$ as given in question are collinear.
Note: One can go wrong while putting angle between two vectors $\overset{\to }{\mathop{a}}\,\text{ and }\overset{\to }{\mathop{b}}\,$which is $0{}^\circ $. He/she may put the angle be $180{}^\circ $ which is wrong.
Need to be careful with angles between $\overset{\wedge }{\mathop{i}}\,,\overset{\wedge }{\mathop{j\ }}\,$and $\overset{\wedge }{\mathop{k}}\,$ during evaluation of $\overset{\to }{\mathop{a}}\,\text{ and }\overset{\to }{\mathop{b}}\,$.