Unit vectors Quiz 1

Question 1

A point exists in a three dimensional coordinate system. Evaluate the unit vector in the direction of the position vector of this point. $$ \ $$ <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/95913f3a-f2bc-4789-9c34-1971db38d6c74380589626858010630.png"/>

Accepted Answer

$$\hat{r}=\dfrac{4\hat{i}+3\hat{j}+2\hat{k}}{\sqrt{29}}$$

Answer

$$\hat{r}=\dfrac{3\hat{i}+\hat{j}+2\hat{k}}{\sqrt{19}}$$

Answer

$$\hat{r}=\dfrac{\hat{i}+3\hat{j}+4\hat{k}}{\sqrt{39}}$$

Answer

None of the above.

Question 2

Imagine two vectors start from the same point and are given as $$\vec{a}=3\hat{i}+4\hat{j}-\hat{k}$$ and $$\vec{b}=2\hat{i}-5\hat{j}+\hat{k}$$ , evaluate the unit vector in the direction of the resultant vector of these two vectors.

Accepted Answer

$$\dfrac{5\hat{i}-\hat{j}}{\sqrt{26}}$$

Answer

$$\dfrac{\hat{i}-5\hat{j}}{\sqrt{26}}$$

Answer

$$\dfrac{5\hat{i}-\hat{j}}{\sqrt{24}}$$

Answer

$$\dfrac{\hat{i}-5\hat{j}}{\sqrt{24}}$$

Question 3

How is concentration of reactant ($${{C}_{r}}$$) and the rate of reaction (k) related:Imagine a vector $$\hat{r}$$ to be a unit vector and the point $$A\left( a,3a,4a ight)$$ to lie in the first quadrant, determine the value of the variable $$a$$ $$ \ $$  <img  src="https://cmds-vedantu.s3.ap-southeast-1.amazonaws.com/prod/question-sets/89826b55-1918-4867-a953-cc3e640f9c352469809482327695192.png"/>

Accepted Answer

$$\dfrac{1}{\sqrt{26}}$$

Answer

$$\sqrt{24}$$

Answer

$$\sqrt{26}$$

Answer

$$\dfrac{1}{\sqrt{24}}$$

Question 4

Rotate the position vector $$4\hat{i}+3\hat{j}$$ by an angle of $${{30}^{\circ }}$$ in the anticlockwise direction.

Accepted Answer

$$\left( 20\sqrt{\dfrac{57-24\sqrt{3}}{955-360\sqrt{3}}} \right)\hat{i}+\left( 5\sqrt{\dfrac{43+24\sqrt{3}}{955-360\sqrt{3}}} \right)\hat{j}$$

Answer

$$\left( 5\sqrt{\dfrac{57-24\sqrt{3}}{955-360\sqrt{3}}} \right)\hat{i}+\left( 20\sqrt{\dfrac{43+24\sqrt{3}}{955-360\sqrt{3}}} \right)\hat{j}$$

Answer

$$\left( \sqrt{\dfrac{24-57\sqrt{3}}{955-360\sqrt{3}}} \right)\hat{i}+\left( 5\sqrt{\dfrac{43+24\sqrt{3}}{955-360\sqrt{3}}} \right)\hat{j}$$

Answer

$$\left( 20\sqrt{\dfrac{57-24\sqrt{3}}{955-360\sqrt{3}}} \right)\hat{i}+\left( \sqrt{\dfrac{24+43\sqrt{3}}{955-360\sqrt{3}}} \right)\hat{j}$$

Question 5

Imagine two position vectors are given as $$\vec{a}=3\hat{i}+4\hat{j}+\hat{k}$$ and $$\vec{b}=\hat{i}+6\hat{j}-\hat{k}$$ , evaluate a vector which is served by a magnitude of $$2$$ units such that this vector is perpendicular to their resultant.

Accepted Answer

$$\dfrac{10}{\sqrt{29}}\hat{i}-\dfrac{4}{\sqrt{29}}\hat{j}	ext{ , }\dfrac{-10}{\sqrt{29}}\hat{i}+\dfrac{4}{\sqrt{29}}\hat{j}$$

Answer

$$\dfrac{-10}{\sqrt{29}}\hat{i}-\dfrac{4}{\sqrt{29}}\hat{j}	ext{ , }\dfrac{10}{\sqrt{29}}\hat{i}+\dfrac{4}{\sqrt{29}}\hat{j}$$

Answer

$$\dfrac{-10}{\sqrt{29}}\hat{i}-\dfrac{4}{\sqrt{29}}\hat{j}	ext{ , }\dfrac{-10}{\sqrt{29}}\hat{i}+\dfrac{4}{\sqrt{29}}\hat{j}$$

Answer

None of these.