Question 11 Mark
Assertion $(A)$ : The vectors :
$\vec{a}=6 \hat{i}+2 \hat{j}-8 \hat{k}, \vec{b}=10 \hat{i}-2 \hat{j}-6 \hat{k}, \vec{c}=4 \hat{i}-4 \hat{j}+2 \hat{k}$
represent the sides of a right angled triangle.
Reason $(R)$ : Three non-zero vectors of which none of two are collinear forms a triangle if their resultant is zero vector or sum of any two vectors is equal to the third.
Answer$\text {Given, } \vec{a}=6 \hat{i}+2 \hat{j}-8 \hat{k}, \vec{b}=10 \hat{i}-2 \hat{j}-6 \hat{k} \text { and } \vec{c}=4 \hat{i}-4 \hat{j}+2 \hat{k}$
$\Rightarrow \vec{a}+\vec{c} =(6 \hat{i}+2 \hat{j}-8 \hat{k})+(4 \hat{i}-4 \hat{j}+2 \hat{k})$
$ =10 \hat{i}-2 \hat{j}-6 \hat{k}=\vec{b}
$Hence, $\vec{a}, \vec{b}$ and $\vec{c}$ are the sides of a triangle.
Let $\theta$ be the angle between $\vec{a}$ and $\vec{c}$, then
$\cos \theta=\frac{\vec{a} \cdot \vec{c}}{|\vec{a}| \cdot|\vec{c}|}=\frac{6 \cdot 4+2(-4)+(-8)(2)}{\sqrt{6^2+2^2+(-8)^2} \sqrt{4^2+(-4)^2+2^2}}$
$=\frac{24-8-16}{\sqrt{104} \sqrt{36}}=0$
$\Rightarrow \cos \theta=\cos 90^{\circ} \Rightarrow \theta=90^{\circ}$
$\therefore \text { Assertion (A) is true. }$
$\therefore \quad$ Assertion $( A )$ is true.
Hence, both Assertion $(A)$ and Reason $(R)$ are true and Reason $(R)$ is the correct explanation of Assertion $(A).$
View full question & answer→Question 21 Mark
Assertion (A): $(\vec{b} \cdot \vec{c}) \vec{a}$ is a scalar quantity.
Reason $(R)$ : Dot product of two vectors is a scalar quantity.
Answer$(\vec{b} \cdot \vec{c}) \vec{a}$ is a vector quantity.
So, Assertion $(A)$ is false but Reason $(R)$ is true.
View full question & answer→Question 31 Mark
Assertion (A): For two hon-zero vectors $\vec{a}$ and $b , \vec{a} \cdot b = b \cdot \vec{a}$.
Reason (R): For two non-zero vectors $\vec{a}$ and $\vec{b}, \vec{a} \times \vec{b}=\vec{b} \times \vec{a}$.
AnswerAssertion (A) is true but reason $( R )$ is false. As, $\vec{a} \times \vec{b}=-\vec{b} \times \vec{a}$.
View full question & answer→Question 41 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The magnitude of the resultant of vectors $\overline{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}$
Reason: The magnitude of a vector can never be negative.
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Answer
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Solution:
$\overline{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\overline{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}},$
Resultant of $\hat{\text{a}}$ and $\hat{\text{b}}$ is $\hat{\text{a}}+\hat{\text{b}}$
$=(2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})+(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})=3\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}}$
$\therefore|\overline{\text{a}}+\overline{\text{b}}|=\sqrt{3^2+3^2+4^2}=\sqrt{9+9+16}=\sqrt{34}$ View full question & answer→Question 51 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The adjacent sides of a parallelogramarealong $\overline{\text{a}}=\hat{\text{i}}+2\hat{\text{j}}$ and $\overline{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}$ The angle between the diagonals is $150^\circ$.
Reason: Two vectors are perpendicular to each other if their dot product is zero.
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Answer
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Solution:
$\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}},\vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}$
Diagonals of the parallelogram arealong $\vec{\text{a}}+\vec{\text{b}}$ and $\vec{\text{a}}-\vec{\text{b}}$
Now, $\vec{\text{a}}+\vec{\text{b}}=(\hat{\text{i}}+2\hat{\text{j}})+(2\hat{\text{i}}+\hat{\text{j}})=3\hat{\text{i}}+3\hat{\text{j}}$
and $\vec{\text{a}}-\vec{\text{b}}=(\hat{\text{i}}+2\hat{\text{j}})-(2\hat{\text{i}}+\hat{\text{j}})=-\hat{\text{i}}+\hat{\text{j}}$
Let $\theta$ be the angle between these vectors, then
$\cos\theta=\frac{(3\text{i}+3\text{j})(\hat{-\text{i}}+\hat{\text{j}})}{\sqrt{9+9}\sqrt{1+1}}=\frac{-3+3}{\sqrt{18}\sqrt{2}}=0$
$\Rightarrow\theta=90^\circ$ View full question & answer→Question 61 Mark
Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: Three points with position vectors $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ are collinear if $\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{c}}\times\vec{\text{a}}=\vec{0}$
Reason: If $\overrightarrow{\text{AB}}.\overrightarrow{\text{AC}}.=0,$ then $\overrightarrow{\text{AB}}\bot\overrightarrow{\text{AC}}.$
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Answer
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Solution:
If $\text{A},\text{B},\text{C}$ are collinear, then $\overrightarrow{\text{AB}}=\overrightarrow{\text{kAC}}$
$\therefore\overrightarrow{\text{AB}}\ \times\overrightarrow{\text{AC}}=\vec{0}$
$\Rightarrow(\vec{\text{b}}-\vec{\text{a}})\times(\vec{\text{c}}-\vec{\text{a}})=0$
$\Rightarrow\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{c}}\times\vec{\text{a}}=\vec{0}$
$\text{i.e}..,\ \vec{\text{a}}\times\vec{\text{b}}+\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{c}}\times\vec{\text{a}}=\vec{0}$ View full question & answer→Question 71 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Let $\vec{\text{a}}$ and $\vec{\text{b}}$ be proper vectors and $\theta$ be the angle between them.
Assertion: $(\vec{\text{a}}\times\vec{\text{b}})^2+(\vec{\text{a}}.\vec{\text{b}})^2\neq(\vec{\text{a}})^2(\vec{\text{b}})^2$
Reason: $\sin^2\theta+\cos^2\theta=0$
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Answer
- Assertion is wrong statement but Reason is correct statement.
Solution:
$(\vec{\text{a}}\times\vec{\text{b}})^2+(\vec{\text{a}}.\vec{\text{b}})^2=|\vec{\text{a}}\times\vec{\text{b}}|^2+(\vec{\text{a}}.\vec{\text{b}})$
$=(\vec{\text{a}}\vec{\text{b}}\sin\theta)^2+(\vec{\text{a}}\vec{\text{b}}\cos\theta)^2=\vec{\text{a}}^2\vec{\text{b}}^2$ View full question & answer→Question 81 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: If $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\vec{0},|\vec{\text{a}}|=3,|\vec{\text{b}}|=4,|\vec{\text{c}}|=5,$ then $\vec{\text{a}}.\vec{\text{b}}+\vec{\text{b}}.\vec{\text{c}}+\vec{\text{c}}.\vec{\text{a}}$ is equal to $-25.$
Reason: If $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\vec0,$ then the$\angle\theta$ between $\vec{\text{b}}$ and $\vec{\text{c}}$ is given by $\cos\theta=\frac{\vec{\text{a}}^2-\vec{\text{b}}^2-\vec{\text{c}}^2}{2\vec{\text{b}}{\vec{\text{c}}}}$
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Answer
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Solution:
We have, $|\vec{\text{a}}|=3,|\vec{\text{b}}|=4,|\vec{\text{c}}|=5, $ and
$\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\vec{0}\Rightarrow(\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}})^2=0$
$\Rightarrow|\vec{\text{a}}|^2+|\vec{\text{b}}|^2+|\vec{\text{c}}|^2+2(\vec{\text{a}}.\vec{\text{b}}+\vec{\text{b}}.\vec{\text{c}}+\vec{\text{c}}.\vec{\text{a}})=0$
$\Rightarrow(3)^2+(4)^2+(5)^2+2(\vec{\text{a}}.\vec{\text{b}}+\vec{\text{b}}.\vec{\text{c}}+\vec{\text{c}}.\vec{\text{a}})=0$
$\Rightarrow\vec{\text{a}}.\vec{\text{b}}+\vec{\text{b}}.\vec{\text{c}}+\vec{\text{c}}.\vec{\text{a}}=\frac{1}{2}[9+16+25]=-\frac{1}{2}(50)=-25$
Now, $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=0\Rightarrow\vec{\text{b}}+\vec{\text{c}}=-\vec{\text{a}}$
$\Rightarrow|\vec{\text{b}}+\vec{\text{c}}|^2=|-\vec{\text{a}}|^2\Rightarrow\vec{\text{b}}^2+\vec{\text{c}}^2+2\vec{\text{b}}.\vec{\text{c}}=\vec{\text{a}}^2$
$\Rightarrow\vec{\text{b}}^2+\vec{\text{c}}^2+2\vec{\text{b}}\vec{\text{c}}\cos\theta=\vec{\text{a}}^2$
$\Rightarrow\cos\theta\frac{\vec{\text{a}}^2-\vec{\text{b}}^2-\vec{\text{c}}^2}{2\vec{\text{b}}\vec{\text{c}}}$ View full question & answer→Question 91 Mark
Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: If $\vec{\text{a}}=2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}},\vec{\text{a}}=-\hat{\text{i}}+3\hat{\text{j}}-4\hat{\text{k}}$ then projection of on .
Reason: Projection of $\vec{\text{a}}$ on $\vec{\text{b}}=\frac{3}{\sqrt{26}}$
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Answer
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Solution:
Projection of $\vec{\text{a}}$ on $\vec{\text{b}}$ $=\frac{\vec{\text{a}}\vec{\text{b}}}{\sqrt{|\vec{\text{b}}|}}$
$\frac{(2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}})(-\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}})}{\sqrt{(-1)^2+(3)^2+(4)^2}}=\frac{-2+9-4}{\sqrt{26}}=\frac{3}{\sqrt{26}}$ View full question & answer→Question 101 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Let $\overline{\text{a}}=\hat{\text{i}}+\hat{\text{j}}=3\hat{\text{k}}$ and $\overline{\text{b}}=\hat{2\text{i}}+\hat{\text{j}}=\hat{\text{k}}$
Assertion: Vectors $\overline{\text{a}}$ and $\overline{\text{b}}$ are perpendicular to each other.
Reason: $\overline{\text{a}}.\overline{\text{b}}=0$
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Answer
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Solution:
$\overline{\text{a}}=\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}},\overline{\text{b}}=\hat{\text{2i}}+\hat{\text{j}}-\hat{\text{k}}$
$\overline{\text{a}}.\overline{\text{b}}=(\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}).(2\hat{\text{i}}+\text{j}+\hat{\text{k}})$
$=1.2+1.1+(-3).1=2+1-3=0$
$\Rightarrow\cos\theta=0\Rightarrow\theta=\frac{\pi}{2}$ View full question & answer→Question 111 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The unit vector in the direction of sum of the vectors $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}$ and $2\hat{\text{j}}+6\hat{\text{k}}$ is $-\frac{1}{7}(3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}).$
Reason: Let $\overline{\text{a}}$ be a non - zero vector, then $\frac{\overline{\text{a}}}{|\overline{\text{a}}|}$ is a unit vector parallel to $\overline{\text{a}}$.
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Answer
- Assertion is wrong statement but Reason is correct statement.
Solution:
Sum of the given vectors
$=(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})+(2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}})+(2\hat{\text{j}}+6\hat{\text{k}})=3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}$
$\therefore$ The unit vector in the direction of the sum of the given vectors
$=\frac{3\hat{\text{i}}+2\hat{\text{j}}+6\hat{{\text{k}}}}{\sqrt{3^2+\text{2}^2+6^2}}=\frac{3\hat{\text{i}}+2\hat{\text{j}}+6\hat{{\text{k}}}}{\sqrt{9+4+36}}=\frac{1}{7}(3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}})$ View full question & answer→Question 121 Mark
Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: If $ (\vec{\text{a}}\times\vec{\text{b}})+(\vec{\text{a}}.\vec{\text{b}})=400$ and $|\vec{\text{a}}|=4,$ then $|\vec{\text{b}}|=9.$
Reason: If $\vec{\text{a}}$ and $\vec{\text{b}}$ are any two vectors, then $(\vec{\text{a}}\times\vec{\text{b}})^2$ is equal to $(\vec{\text{a}})^2(\vec{\text{b}})^2-(\vec{\text{a}}.\vec{\text{b}})^2.$
- Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- Assertion is correct statement but Reason is wrong statement.
- Assertion is wrong statement but Reason is correct statement.
Answer
- Assertion is wrong statement but Reason is correct statement.
Solution:
$(\vec{\text{a}}\times\vec{\text{b}})^2+(\vec{\text{a}}.\vec{\text{b}})^2=400,|\vec{\text{a}}|=4$
We know that,
$(\vec{\text{a}}\times\vec{\text{b}})^2+(\vec{\text{a}}.\vec{\text{b}})^2=|\vec{\text{a}}|^2|\vec{\text{b}}|^2$
$=400=(4)^2|\vec{\text{b}}|^2\Rightarrow16|\vec{\text{b}}|^2=400$
$\Rightarrow|\vec{\text{b}}|^2=25\Rightarrow|\vec{\text{b}}|=5$
Hence, Assertion is wrong.
$(\vec{\text{a}}\times\vec{\text{b}})^2+(\vec{\text{a}}.\vec{\text{b}})^2=|\vec{\text{a}}\times\vec{\text{b}}|^2+(\vec{\text{a}}.\vec{\text{b}})^2$
$=(\vec{\text{a}}\vec{\text{b}}\sin\theta)^2+(\vec{\text{a}}\vec{\text{b}}\cos\theta)^2=\vec{\text{a}}^2\vec{\text{b}}^2$ View full question & answer→Question 131 Mark
Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: In $\triangle\text{ABC},\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{CA}}=0.$
Reason: If $\overrightarrow{\text{OA}}=\overrightarrow{\text{a}},\overrightarrow{\text{OB}},\overrightarrow{\text{b}},$ then $\overrightarrow{\text{AB}}=\overrightarrow{\text{a}}+\overrightarrow{\text{b}}.$
- Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
- Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
- Assertion is correct but Reason is incorrect.
- Both Assertion and Reason are incorrect.
Answer
- Assertion is correct but Reason is incorrect.
Solution:
In $\triangle\text{ABC},\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{AC}}=-\overrightarrow{\text{CA}}$
$\Rightarrow\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{CA}}=\overrightarrow{0}$
$\overrightarrow{\text{OA}}+\overrightarrow{\text{AB}}=\overrightarrow{\text{OB}}$ is the triangle law of addition. View full question & answer→Question 141 Mark
Assertion $(A) :$ The adjacent sides of a parallelogram are along $\vec{a}=\hat{i}+2 \hat{j}$ and $\vec{b}=2 \hat{i}+\hat{j}$. The angle between the diagonals is $150^{\circ}$.
Reason $(R) :$ Two vectors are perpendicular to each other if their dot product is zero.
Answer$(d) : \vec{a}=\hat{i}+2 \hat{j}, \vec{b}=2 \hat{i}+\hat{j}$
Diagonals of the parallelogram are along $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$.
Now, $\vec{a}+\vec{b}=(\hat{i}+2 \hat{j})+(2 \hat{i}+\hat{j})=3 \hat{i}+3 \hat{j}$
and $\vec{a}-\vec{b}=(\hat{i}+2 \hat{j})-(2 \hat{i}+\hat{j})=-\hat{i}+\hat{j}$
Let $\theta$ be the angle between these vectors, then
$\cos \theta=\frac{(3 \hat{i}+3 \hat{j}) \cdot(-\hat{i}+\hat{j})}{\sqrt{9+9} \sqrt{1+1}}=\frac{-3+3}{\sqrt{18} \sqrt{2}}=0$
$\Rightarrow \theta=90^{\circ}$
Hence, assertion is false and reason is true.
View full question & answer→Question 151 Mark
Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors and $\theta$ be the angle between then.
Assertion (A) : $(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2 \neq|\vec{a}|^2|\vec{b}|^2$
Reason (R) : $\sin ^2 \theta+\cos ^2 \theta=1$
Answer(d) :
$
\begin{aligned}
(\vec{a} \times \vec{b})^2 & +(\vec{a} \cdot \vec{b})^2=|\vec{a} \times \vec{b}|^2+(\vec{a} \cdot \vec{b})^2 \\
& =(|\vec{a}|| \vec{b} \mid \sin \theta)^2+(|\vec{a}||\vec{b}| \cos \theta)^2=|\vec{a}|^2|\vec{b}|^2
\end{aligned}
$
Hence, Assertion is false.
But $\sin ^2 \theta+\cos ^2 \theta=1$
Hence, reason is true.
View full question & answer→Question 161 Mark
Assertion $(A) :$ If $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0},|\vec{a}|=3$, $|\vec{b}|=4,|\vec{c}|=5$, then $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ is equal to $-25 .$
Reason $(R) :$ If $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, then the angle $\theta$ between $\vec{b}$ and $\vec{c}$ is given by
$\cos \theta=\frac{|\vec{a}|^2-|\vec{b}|^2-|\vec{c}|^2}{2|\vec{b}||\vec{c}|^2} .$
Answer$(b) :$ We have, $|\vec{a}|=3,|\vec{b}|=4,|\vec{c}|=5$ and
$\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0} \Rightarrow(\vec{a}+\vec{b}+\vec{c})^2=0$
$\Rightarrow|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0$
$\Rightarrow(3)^2+(4)^2+(5)^2+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0$
$\Rightarrow \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}=-\frac{1}{2}[9+16+25]=-\frac{1}{2}(50)=-25$
$\text { Now, } \vec{a}+\vec{b}+\vec{c}=\overrightarrow{0} \Rightarrow \vec{b}+\vec{c}=-\vec{a}$
$\Rightarrow(\vec{b}+\vec{c})^2=(-\vec{a})^2 \Rightarrow|\vec{b}|^2+|\vec{c}|^2+2 \vec{b} \cdot \vec{c}=|\vec{a}|^2$
$\Rightarrow|\vec{b}|^2+|\vec{c}|^2+2|\vec{b}||\vec{c}| \cos \theta=|\vec{a}|^2$
$\Rightarrow \cos \theta=\frac{|\vec{a}|^2-|\vec{b}|^2-|\vec{c}|^2}{2|\vec{b}||\vec{c}|}$
Hence, both assertion and reason are true but reason is not the correct explanation of assertion.
View full question & answer→Question 171 Mark
Let $\vec{a}=\hat{i}+\hat{j}-3 \hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}+\hat{k}$.
Assertion $(A):$ Vectors $\vec{a}$ and $\vec{b}$ are perpendicular to each other.
Reason $(R) : \vec{a} \cdot \vec{b}=0$
Answer$\text { (a): } \vec{a}=\hat{i}+\hat{j}-3 \hat{k}, \vec{b}=2 \hat{i}+\hat{j}+\hat{k}$
$\vec{a} \cdot \vec{b} =(\hat{i}+\hat{j}-3 \hat{k}) \cdot(2 \hat{i}+\hat{j}+\hat{k})$
$ =1 \cdot 2+1 \cdot 1+(-3) \cdot 1=2+1-3=0$
$\Rightarrow \cos \theta=0 \Rightarrow \theta=\frac{\pi}{2}$
Hence, $\vec{a}$ and $\vec{b}$ are perpendicular to each other.
Hence, both assertion and reason are true and reason is the correct explanation of assertion.
View full question & answer→Question 181 Mark
Assertio$n (A) :$ Three points with position vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are collinear if $\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=\overrightarrow{0}$
Reason $(R):$ If $\overrightarrow{A B} \cdot \overrightarrow{A C}=0$, then $\overrightarrow{A B} \perp \overrightarrow{A C}$.
Answer(b) : If $A, B, C$ are collinear, then $\overrightarrow{A B}=k \overrightarrow{A C}$
$\therefore \overrightarrow{A B} \times \overrightarrow{A C}=\overrightarrow{0} \Rightarrow(\vec{b}-\vec{a}) \times(\vec{c}-\vec{a})=\overrightarrow{0}$
$\Rightarrow \vec{b} \times \vec{c}+\vec{a} \times \vec{b}+\vec{c} \times \vec{a}=\overrightarrow{0} \text { i.e., } \vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=\overrightarrow{0}$
Hence, both assertion and reason are true but reason is not the correct explanation of assertion.
View full question & answer→Question 191 Mark
Assertion $(A) :$ If the points $\vec{P}=(\vec{a}+\vec{b}-\vec{c})$, $\vec{Q}=(2 \vec{a}+\vec{b})$ and $\vec{R}=(\vec{b}+t \vec{c})$ are collinear, where $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors, then the value of $t$ is $-2 .$
Reason $(R) :$ If $P, Q, R$ are collinear, then
$\overrightarrow{P Q} \| \overrightarrow{P R} \text { or } \overrightarrow{P Q}=\lambda \overrightarrow{P R}, \lambda \in R$
Answer$(a) :$ If $P, Q, R$ are collinear, then $\overrightarrow{P Q} \| \overrightarrow{P R}$ or $\overrightarrow{P Q}=\lambda \overrightarrow{P R}, \lambda \in R$
$\Rightarrow \quad(2 \vec{a}+\vec{b})-(\vec{a}+\vec{b}-\vec{c})=\lambda[(\vec{b}+t \vec{c})-(\vec{a}+\vec{b}-\vec{c})]$
$\Rightarrow \quad(\vec{a}+\vec{c})=\lambda[-\vec{a}+(t+1) \vec{c}]$
$\Rightarrow \vec{a}+\vec{c}=-\lambda \vec{a}+\lambda(t+1) \vec{c}$
On comparing, we get
$-\lambda=1$
$\Rightarrow \lambda=-1$
$\text { and } \lambda(t+1)=1 \Rightarrow-(t+1)=1$
$\Rightarrow-t-1=1$
$\Rightarrow t=-2$
Hence, both assertion are true and reason is the correct explanation of assertion.
View full question & answer→Question 201 Mark
Assertion $(A) :$ The magnitude of resultant of vectors $\vec{a}=2 \hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}$ is $\sqrt{34}$.
Reason $(R) :$ The magnitude of a vector can never be negative.
Answer$(b) : \vec{a}=2 \hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}$
Addition of $\vec{a}$ and $\vec{b}$ is $\vec{a}+\vec{b}$
$\vec{a}+\vec{b}=(2 \hat{i}+\hat{j}+\hat{k})+(\hat{i}+2 \hat{j}+3 \hat{k})=3 \hat{i}+3 \hat{j}+4 \hat{k}$
$\therefore|\vec{a}+\vec{b}|=\sqrt{3^2+3^2+4^2}=\sqrt{9+9+16}=\sqrt{34}$
Also, the magnitude of a vector can never be negative. Hence, both Assertion and Reason are correct but Reason is not the correct explanation of Assertion.
View full question & answer→Question 211 Mark
Assertion $(A) :$ If $(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=400$ and $|\vec{a}|=4$, then $|\vec{b}|=9$.
Reason $(R) :$ If $\vec{a}$ and $\vec{b}$ are any two vectors, then $(\vec{a} \times \vec{b})^2$ is equal to $|\vec{a}|^2|\vec{b}|^2-(\vec{a} \cdot \vec{b})^2$.
Answer$(d): (\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=400,|\vec{a}|=4$
Now, $(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=|\vec{a}|^2|\vec{b}|^2$
$\Rightarrow 400=(4)^2|\vec{b}|^2 \Rightarrow 16|\vec{b}|^2=400$
$\Rightarrow|\vec{b}|^2=25 \Rightarrow|\vec{b}|=5$
Hence, assertion is false.
$(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=|\vec{a} \times \vec{b}|^2+(\vec{a} \cdot \vec{b})^2$
$=(|\vec{a}||\vec{b}| \sin \theta)^2+(|\vec{a}||\vec{b}| \cos \theta)^2=|\vec{a}|^2|\vec{b}|^2$
$\Rightarrow(\vec{a} \times \vec{b})^2=|\vec{a}|^2|\vec{b}|^2-(\vec{a} \cdot \vec{b})^2$
Hence, reason is true.
View full question & answer→Question 221 Mark
Assertion (A) : The unit vector in the direction of sum of the vectors $\hat{i}+\hat{j}+\hat{k}, 2 \hat{i}-\hat{j}-\hat{k}$ and $2 \hat{j}+6 \hat{k}$ is $\frac{1}{7}(3 \hat{i}-2 \hat{j}+6 \hat{k})$.
Reason (R) : Let $\vec{a}$ be a non-zero vector, then $\frac{\vec{a}}{|\vec{a}|}$ is a unit vector parallel to $\vec{a}$.
Answer(d) : Sum of the given vectors
$
=(\hat{i}+\hat{j}+\hat{k})+(2 \hat{i}-\hat{j}-\hat{k})+(2 \hat{j}+6 \hat{k})=3 \hat{i}+2 \hat{j}+6 \hat{k}
$
$\therefore \quad$ The unit vector in the direction of the sum of the given vectors is given by
$
=\frac{3 \hat{i}+2 \hat{j}+6 \hat{k}}{\sqrt{3^2+2^2+6^2}}=\frac{3 \hat{i}+2 \hat{j}+6 \hat{k}}{\sqrt{9+4+36}}=\frac{1}{7}(3 \hat{i}+2 \hat{j}+6 \hat{k})
$
Hence, Assertion is false.
Also, $\frac{\vec{a}}{|\vec{a}|}$ is a unit vector which is parallel to $\vec{a}$.
Hence, Reason is true.
View full question & answer→Question 231 Mark
Assertion (A) : The projection of the vector $3 \hat{i}-\hat{j}-2 \hat{k}$ on the vector $\hat{i}+2 \hat{j}-3 \hat{k}$ is $\frac{7}{\sqrt{14}}$.
Reason (R) : The projection of a vector $\vec{a}$ on another vector $\vec{b}$ is $\frac{(\vec{a} \cdot \vec{b})}{|\vec{b}|}$.
Answer(a) : Required projection $=\frac{(3 \hat{i}-\hat{j}-2 \hat{k}) \cdot(\hat{i}+2 \hat{j}-3 \hat{k})}{\sqrt{1^2+2^2+(-3)^2}}$
$
\frac{3-2+6}{\sqrt{1+4+9}}=\frac{7}{\sqrt{14}}
$
Also, projection of vector $\vec{a}$ on $\vec{b}=(\vec{a} \cdot \hat{b})=\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\right)$
Hence, both assertion and reason are true and reason is the correct explanation of assertion.
View full question & answer→