Question

What vector must be added to the two vectors i- 2j + 2k and 2i+j-k that the resultant may be a unit vector along x axis

Answer 1

Answer:

The vector that must be added is $-2\hat i+\hat j-\hat k$

Explanation:

Unit vector along x-axis is $\hat i$

Let the vector added is $a_1\hat i+a_2\hat j+a_3\hat k$

Therefore

$(\hat i-2\hat j+2\hat k)+(2\hat i+\hat j-\hat k)+(a_1\hat i+a_2\hat j+a_3\hat k)=\hat i$

$\implies 3\hat i-\hat j+\hat k+a_1\hat i+a_2\hat j+a_3\hat k=\hat i$

$\implies (3+a_1)\hat i+(a_2-1)\hat j+(a_3+1)\hat k=\hat i$

Comparing both the sides

$a_1+3=1\implies a_1=-2$

$a_2-1=0\implies a_2=1$

$a_3+1=0\implies a_3=-1$

Therefore, the vector that must be added is

$-2\hat i+\hat j-\hat k$

Hope this answer is helpful.

Answer 2

Answer:

$\vec{C}\ =\ -2i\ +\ j\ -\ k$

Explanation:

Given,

* first vector = $\vec{A}\ =\ i\ -\ 2j\ +\ 2k$

*second vector = $\vec{B}\ =\ 2i\ +\ j\ -\ k$

Let \vec C be the third vector,

Resultant of the sum of the three vectors = unit vector along x-direction = i

$\vec{A}\ +\ \vec{B}\ +\ \vec{C}\ =\ i\\\Rightarrow \vec{C}\ =\ i\ -\ \vec{A}\ -\ \vec{B}\\\Rightarrow \vec{C}\ =\ i\ -\ (i\ -\ 2j\  +\ 2k)\ -\ (2i\ +\ j\ -\ k)\\\Rightarrow \vec{C}\ =\ i\ -\ i\ +2j\ -\ 2k\ -\ 2i\ -j\ +\ k\\\Rightarrow \vec{C}\ =\ -2i\ +\ j\ +\ -k$