Question

Find the unit vector in the direction of the sum of the vectors, \(\vec{a}\)=2\(\hat{i}\)+7\(\hat{j}\) and \(\vec{b}\)=\(\hat{i}\)-9\(\hat{j}\).

(a) \(\frac{3}{\sqrt{11}} \hat{i}-\frac{2}{\sqrt{11}} \hat{j}\)

(b) \(\frac{2}{\sqrt{13}} \hat{i}-\frac{3}{\sqrt{13}} \hat{j}\)

(c) –\(\frac{3}{\sqrt{11}} \hat{i}+\frac{2}{\sqrt{13}} \hat{j}\)

(d) \(\frac{3}{\sqrt{13}} \hat{i}-\frac{2}{\sqrt{13}} \hat{j}\)

This question was posed to me in exam.

This question is from Addition of Vectors in chapter Vector Algebra of Mathematics – Class 12

Answer 1

The correct answer is (d) \(\frac{3}{\sqrt{13}} \hat{i}-\frac{2}{\sqrt{13}} \hat{j}\)

The explanation: Given that, \(\vec{a}\)=2\(\hat{i}\)+7\(\hat{j}\) and \(\vec{b}\)=\(\hat{i}\)-9\(\hat{j}\)

The sum of the two vectors will be

\(\vec{a}+\vec{b}\)=(2\(\hat{i}\)+7\(\hat{j}\))+(\(\hat{i}\)-9\(\hat{j}\))

=(2+1) \(\hat{i}\)+(7-9)\(\hat{j}\)

=3\(\hat{i}\)-2\(\hat{j}\)

The unit vector in the direction of the sum of the vectors is

\(\frac{1}{|\vec{a}+\vec{b}|}  (\vec{a}+\vec{b})=\frac{3\hat{i}-2\hat{j}}{\sqrt{3^2+(-2)^2}}=\frac{3\hat{i}-2\hat{j}}{\sqrt{13}}=\frac{3}{1\sqrt{3}} \hat{i}-\frac{2}{\sqrt{13}}\hat{j}\)