Question

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Answer 1

Given :

• $\vec{A} = -2\hat{\imath} + 3\hat{\jmath}+\hat{k}$

• $\vec{B} = \hat{\imath} + 2\hat{\jmath}-4\hat{k}$

To find :

• Unit vector in the direction of resultant vector of vectors $\vec{A}$ and $\vec{B}$

Solution :

Resultant vector is the sum of given vectors

$\vec{R} = \vec{A}+\vec{B}$

${ \vec{R} = (-2\hat{\imath} + 3\hat{\jmath}+\hat{k} )+(\hat{\imath} + 2\hat{\jmath}-4\hat{k})}$

${ \vec{R} = -2\hat{\imath} + 3\hat{\jmath}+\hat{k} +\hat{\imath} + 2\hat{\jmath}-4\hat{k}}$

${ \vec{R} = -\hat{\imath} + 5\hat{\jmath}-3\hat{k}}$

Now, magnitude of this vector is given by,

${ | \vec{R} | = \sqrt{ {x}^{2} + {y}^{2} + {z}^{2} } }$

${ | \vec{R} | = \sqrt{ { 1}^{2} + {5}^{2} + {3}^{2} } }$

${ | \vec{R} | = \sqrt{ { 1}+ 25 + 9 } }$

${ | \vec{R} | = \sqrt{ 35 } }$

Required unit vector is given by,

$\rm Unit\: Vector= \dfrac{\vec{R}}{|\vec{R}|}$

$\rm Unit\: Vector= \dfrac{-\hat{\imath} + 5\hat{\jmath}-3\hat{k}}{ \sqrt{35} }$

Hence option (C) is correct