# 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$$\hat{j}$$.

Category: QuestionsFind 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$$\hat{j}$$.
Editor">Editor Staff asked 11 months ago

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
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Editor">Editor Staff answered 11 months ago

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

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