Question

A vector Č when added to two vectors Ā=(î – 2ị + 4k) and B=(3î + 5j – 7k) gives a unit vector
along x - axis. Find the vector C.​

Answer 1

Answer:

We are given :

A = (î - 2j + 4k) and B = (3î + 5j -7k).

Adding both : A + B = (î - 2j + 4k) + (3î + 5j - 7k)

⇒ A + B = (4î + 3j - 3k)

The unit vector along the x-axis is : î.

Hence, Required vector = C = î - (4î + 3j - 3k)

⇒ C = -3î - 3j + 3k

EXTRA INFORMATION –

What is a vector ?

A quantity which has both a magnitude and direction is called a vector.

What is unit vector ?

A unit vector is a vector of unit magnitude and points in a particular direction. It has no dimension and no unit. It specifically points in a direction only. The unit vector of x, y and z axes are î, j and k respectively.

Answer 2

$\huge\mathcal{\underbrace{\red{QUESTION:-}}}$

✨ A vector C when added to the resultant of the two vectors $\rm{\vec{A}\:=\:\hat{i}\:-\:2\hat{j}\:+\:4\hat{k}\:}$ and $\rm{\vec{B}\:=\:3\hat{i}\:+\:5\hat{j}\:-\:7\hat{k}\:}$ gives a unit vector along x-axis . Find the vector C .

$\huge\mathcal{\underbrace{\red{ANSWER:-}}}$

$\mathcal{\gray{\underbrace{\blue{GIVEN:-}}}}$

• A vector C when added to the resultant of the two vectors $\rm{\vec{A}\:=\:\hat{i}\:-\:2\hat{j}\:+\:4\hat{k}\:}$ and $\rm{\vec{B}\:=\:3\hat{i}\:+\:5\hat{j}\:-\:7\hat{k}\:}$ gives a unit vector along x-axis .

$\mathcal{\gray{\underbrace{\blue{TO\: FIND:-}}}}$

• Find the vector C .

$\mathcal{\gray{\underbrace{\blue{SOLUTION:-}}}}$

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

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

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

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

$\rm{\green{\implies\:\vec{R}\:=\:4\hat{i}\:+\:3\hat{j}\:-\:3\hat{k}\:}}$

According to the question,

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

☞ $\rm\red{\hat{i}}$ represents along x-axis.

☞ $\rm\red{\hat{j}}$ represents along y-axis .

☞ $\rm\red{\hat{k}}$ represents along z-axis .

Hence,

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

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

$\rm{\implies\:\vec{C}\:=\:1\hat{i}\:-\:[4\hat{i}\:+\:3\hat{j}\:-\:3\hat{k}]\:}$

$\rm{\implies\:\vec{C}\:=\:1\hat{i}\:-\:4\hat{i}\:-\:3\hat{j}\:+\:3\hat{k}\:}$

$\rm{\green{\implies\:\vec{C}\:=\:-\:3\hat{i}\:-\:3\hat{j}\:+\:3\hat{k}\:}}$

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