Question 2, Exercise 3.2

Solutions of Question 2 of Exercise 3.2 of Unit 03: Vectors. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.

Find unit vector having the same direction as the vector $3\hat{i}.$

Let $$\overset{\scriptscriptstyle\rightharpoonup}{a}=3\hat{i}$$ Then $$|\overset{\scriptscriptstyle\rightharpoonup}{a}|=\sqrt{{{(3)}^{2}}}=3$$ Now we know that $$\hat{a}=\dfrac{{\overset{\scriptscriptstyle\rightharpoonup}{a}}}{|\overset{\scriptscriptstyle\rightharpoonup}{a}|}=\dfrac{3\hat{i}}{3}=\hat{i}$$ This is the required unit vector.

Find unit vector having the same direction as the vector $3\hat{i}-4\hat{j}.$

Let $$\overset{\scriptscriptstyle\rightharpoonup}{a}=3\hat{i}-4\hat{j}$$ Then $$|\overset{\scriptscriptstyle\rightharpoonup}{a}|=\sqrt{{{(3)}^{2}}+{{(-4)}^{2}}}=5$$ Now we know that $$\hat{a}=\dfrac{{\overset{\scriptscriptstyle\rightharpoonup}{a}}}{|\overset{\scriptscriptstyle\rightharpoonup}{a}|}=\dfrac{3\hat{i}-4\hat{j}}{5}=\dfrac{3}{5}\hat{i}-\dfrac{4}{5}\hat{j}$$ Which is the required unit vector.

Find unit vector having the same direction as the vector $\hat{i}+\hat{j}-2\hat{k}.$

Let $$\overset{\scriptscriptstyle\rightharpoonup}{a}=\hat{i}+\hat{j}-2\hat{k}$$
Then
$$|\overset{\scriptscriptstyle\rightharpoonup}{a}|=\sqrt{(1)^2+(1)^2+(-2)^2}=\sqrt{6}$$
Now we know that
$$\hat{a}=\dfrac{{\overset{\scriptscriptstyle\rightharpoonup}{a}}}{|\overset{\scriptscriptstyle\rightharpoonup}{a}|}=\dfrac{\hat{i}+\hat{j}-2\hat{k}}{\sqrt{6}}=\dfrac{1}{\sqrt{6}}\hat{i}+\dfrac{1}{\sqrt{6}}\hat{j}-\dfrac{2}{\sqrt{6}}\hat{k}$$
Which is the unit vector.

Find unit vector having the same direction as the vector $\dfrac{\sqrt{3}}{2}\hat{i}-\dfrac{1}{2}\hat{j}.$

Let $$\overset{\scriptscriptstyle\rightharpoonup}{a}=\dfrac{\sqrt{3}}{2}\hat{i}-\dfrac{1}{2}\hat{j}$$
Then
$$|\overset{\scriptscriptstyle\rightharpoonup}{a}|=\sqrt{{{(\dfrac{\sqrt{3}}{2})}^{2}}+{{(\dfrac{1}{2})}^{2}}}=1$$
Now we know that
$$\hat{a}=\dfrac{{\overset{\scriptscriptstyle\rightharpoonup}{a}}}{|\overset{\scriptscriptstyle\rightharpoonup}{a}|}=\dfrac{\dfrac{\sqrt{3}}{2}\hat{i}-\dfrac{1}{2}\hat{j}}{1}=\dfrac{\sqrt{3}}{2}\hat{i}-\dfrac{1}{2}\hat{j}$$
Which is the unit vector.