Question

What vector when added to $(2 \hat{i} – 2 \hat{j} + \hat{k})$ and $(2 \hat{i} - \hat{k})$ will give a unit vector along negative y-axis?

Answer 1

Hint: We have given two vectors. We have to find another vector which when added to the sum of given vectors then it will give a unit vector along the negative y-axis. So, we will find out that vector after comparing the coefficients of the basis vector.

Complete step-by-step solution:
Let $\vec{A} = (2 \hat{i} – 2 \hat{j} + \hat{k})$ and $\vec{B} = (2 \hat{i} - \hat{k})$.
We need to find a vector $\vec{C}$ which when added to $\vec{A} + \vec{B}$ gives a unit vector along the negative y-axis.
So, we first take the sum of vectors $\vec{A}$ and $\vec{B}$.
$\vec{A} + \vec{B} = (2 \hat{i} – 2 \hat{j} + \hat{k}) + (2 \hat{i} - \hat{k}) $
We get,
$\vec{A} + \vec{B} = 4 \hat{i} – 2 \hat{j} $
 Let $\vec{C} = (x \hat{i} + y \hat{j} + z\hat{k})$
Now add vector $\vec{C}$ to $\vec{A} + \vec{B}$
$\vec{A} + \vec{B}+ \vec{C} = ( 4 \hat{i} – 2 \hat{j}) + (x \hat{i} + y \hat{j} + z\hat{k}) $
Now according to the question,
 $( 4 \hat{i} – 2 \hat{j}) + (x \hat{i} + y \hat{j} + z\hat{k}) = -\hat{j}$
$( 4+x) \hat{i} +(y– 2) \hat{j} + z\hat{k} = -\hat{j}$
Compare the coefficient of i, j and k.
$(4 + x) = 0 \implies x =-4$
$y -2= -1 \implies y =1$
$z = 0$
Now vector $\vec{C}$ will be,
$\vec{C} = -4 \vec{i} + \hat{j}$

Note:A vector contains both a magnitude and a direction. Geometrically, we can imagine a vector as a pointed line segment, whose length is the vector's magnitude, and with an arrow showing the direction. The direction of the vector is from its end to its head.