Question

A vector x, when added to the resultant of the vectors A = 3î - 59 + 7k and ß = 2î +49-3k gives a unit vector
along Y-axis. Find the vector x.​

Answer 1

Correct Question:-

A vector x, when added to the resultant of the vectors $\large\rm{ A = 3 \hat{i} - 5 \hat{j} + 7 \hat{k}}$ and B = 2î +4j-3k gives a unit vector

along Y-axis. Find the vector x.

Given that:

A = 3î - 5j + 7k

B = 2î + 4j - 3k

Answer:-

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

$\large\rm { \vec{R} = 5 \hat{i} - \hat{j} + 4 \hat{k}}$

$\large\rm { \vec{x} + \vec{R} = j}$

$\large\rm { \vec{x} + 5 \hat{i} - \hat{j} + 4 \hat{k} = \hat{j}}$

$\large\rm { \vec{x} = -5 \hat{i} + 2 \hat{j} - 4 \hat{k}}$

Answer 2

$\huge\mathsf{\underbrace{\red{QUESTION:-}}} </p><p>$

✨ A vector C when added to the resultant of the two vectors

$\green{\rm{\vec{A}\:=\:\hat{i}\:-\:2\hat{j}\:+\:4\hat{k}\:} </p><p>}$

And

$\blue{\rm{\vec{B}\:=\:3\hat{i}\:+\:5\hat{j}\:-\:7\hat{k}\:} </p><p>}</p><p>$

Gives a unit vector along x-axis . Find the vector C .

$\huge\mathsf{\underbrace{\red{ANSWER:-}}} </p><p>$

$\mathsf{\gray{\underbrace{\blue{GIVEN:-}}}} </p><p>$

A vector C when added to the resultant of the two vectors

$\bf{\vec{A}\:=\:\hat{i}\:-\:2\hat{j}\:+\:4\hat{k}\:} </p><p>$

And

$\pink{ \rm{\vec{B}\:=\:3\hat{i}\:+\:5\hat{j}\:-\:7\hat{k}\:} }</p><p>$

Gives a unit vector along x-axis .

$\mathsf{\gray{\underbrace{\blue{TO\: FIND:-}}}} </p><p>$

Find the vector C .

$\huge\mathsf{\gray{\underbrace{\blue{SOLUTION:-}}}} </p><p>$

Let, R be the resultant of vector A and vector B .

$\red{\rm{\vec{R}\:=\:[\hat{i}\:-\:2\hat{j}\:+\:4\hat{k}]\:+\:[3\hat{i}\:+\:5\hat{j}\:-\:7\hat{k}]\:} </p><p>}$

$\rm{\implies\:\vec{R}\:=\:\hat{i}\:-\:2\hat{j}\:+\:4\hat{k}\:+\:3\hat{i}\:+\:5\hat{j}\:-\:7\hat{k}\:}</p><p>$

$\green{\rm{\implies\:\vec{R}\:=\:\hat{i}\:+\:3\hat{i}\:-\:2\hat{j}\:+\:5\hat{j}\:+\:4\hat{k}\:-\:7\hat{k}\:}</p><p> }$

$\rm{\blue{\implies\:\vec{R}\:=\:4\hat{i}\:+\:3\hat{j}\:-\:3\hat{k}\:}}</p><p>$

According to the question,

☯️The sum of vector C and vector R gives a unit vector along x-axis .

$\pink\checkmark \:\rm\red{\hat{i}}$

represents along x-axis.

$\pink\checkmark \: \rm\red{\hat{j}}$

Represents along y-axis .

$\pink \checkmark \: \rm\red{\hat{k}}$

Represents along z-axis .

Hence,

$\orange\checkmark\:\rm{\purple{\vec{C}\:+\:\vec{R}\:=\:1\hat{i}\:}}$

$\blue{\rm{\implies\:\vec{C}\:=\:1\hat{i}\:-\:\vec{R}\:}}$

$\green{\bf{\implies\:\vec{C}\:=\:1\hat{i}\:-\:[4\hat{i}\:+\:3\hat{j}\:-\:3\hat{k}]\:}}</p><p>$

$\orange{\rm{\implies\:\vec{C}\:=\:1\hat{i}\:-\:4\hat{i}\:-\:3\hat{j}\:+\:3\hat{k}\:}}</p><p>$

$\rm{\red{\implies\:\vec{C}\:=\:-\:3\hat{i}\:-\:3\hat{j}\:+\:3\hat{k}\:}}</p><p>$

$\rm\pink{\therefore} The \: vector \: C \: is “\rm{\green{-\:3\hat{i}\:-\:3\hat{j}\:+\:3\hat{k}\:}}$