Sample QuestionsVector Algebra questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $\vec{a}$ is a nonzero vector of magnitude ' $a$ ' and $\lambda$ a nonzero scalar, then $\lambda \vec{a}$ is unit vector if :
- A
$\lambda=1$
- B
$\lambda=-1$
- C
$a=|\lambda|$
- ✓
$a=\frac{1}{|\lambda|}$
Answer: D.
View full solution →If the magnitudes of two vectors $\vec{a}$ and $\vec{b}$ are $\sqrt{3}$ and 2 respectively and $\vec{a} \cdot \vec{b}=\sqrt{6}$. Then the angle between $\vec{a}$ and $\vec{b}$ is:
- A
$\frac{\pi}{2}$
- B
$\frac{\pi}{3}$
- C
$\frac{\pi}{6}$
- ✓
$\frac{\pi}{4}$
Answer: D.
View full solution →The value of $\hat{i} \cdot(\hat{j} \times \hat{k})+\hat{j} \cdot(\hat{i} \times \hat{k})+k \cdot(\hat{i} \times \hat{j})$ is:
Answer: C.
View full solution →The unit vector along the vector $\vec{a}=-2 \hat{i}+3 \hat{j}-\hat{k}$ is :
- A
$\frac{2 \hat{i}}{\sqrt{14}}-\frac{3 \hat{j}}{\sqrt{14}}+\frac{\hat{k}}{\sqrt{14}}$
- B
$\frac{2 \hat{i}}{\sqrt{14}}-\frac{3 \hat{j}}{\sqrt{14}}-\frac{\hat{k}}{\sqrt{14}}$
- C
$\frac{2 \hat{i}}{\sqrt{14}}+\frac{3 \hat{j}}{\sqrt{14}}-\frac{\hat{k}}{\sqrt{14}}$
- ✓
$\frac{-2 \hat{i}}{\sqrt{14}}+\frac{3 \hat{j}}{\sqrt{14}}-\frac{\hat{k}}{\sqrt{14}}$
Answer: D.
View full solution →The magnitude of the vector $\frac{1}{\sqrt{3}} \hat{i}+\frac{1}{\sqrt{3}} \hat{j}-\frac{1}{\sqrt{3}} \hat{k}$ is:
Answer: B.
View full solution →Find the projection of the vector $\vec{a}=2 \hat{i}+3 \hat{j}+2 \hat{k}$ on the vector $\vec{b}=\hat{i}+2 \hat{j}+\hat{k}$.
View full solution →Find the position vector of the mid-point of the vector joining the points $P (2,3,4)$ and $Q (4,1,-2)$.
View full solution →The initial point of a vector is $(2,1)$ and the terminal point is $(-5,7)$, find the vectors components of this vector.
View full solution →The vector directed from A to B and joining the points $A (1,2,2)$ and $B (2,3,1)$ is :
View full solution →In triangle OAC if B is the mid-point of side AC and $\overrightarrow{ OA }=\vec{a}$ and $\overrightarrow{ OB }=\vec{b}$, then what will be $\overrightarrow{ OC }$ ?
View full solution →Find the area of the parallelogram whose adjacent sides are determined by the vectors $\vec{a}=\hat{i}-\hat{j}+3 \hat{k}$ and $\vec{b}=2 \hat{i}-7 \hat{j}+\hat{k}$.
View full solution →Evaluate the product $(3 \vec{a}-5 \vec{b}) \cdot(2 \vec{a}+7 \vec{b})$
View full solution →If $|\vec{a}|=10,|\vec{b}|=2$ and $\vec{a} \cdot \vec{b}=12$, then find the value of $|\vec{a} \times \vec{b}|$.
View full solution →If two vectors $\vec{a}$ and $\vec{b}$ are such that $|\vec{a}|=2,|\vec{b}|=3$ and $\vec{a} \cdot \vec{b}=4$, then find $|\vec{a}-\vec{b}|$.
View full solution →Find the unit vector along the sum of the vectors $\vec{a}=2 \hat{i}+2 \hat{j}-5 \hat{k}$ and $\vec{b}=2 \hat{i}+2 \hat{j}+\hat{k}$.
View full solution →If two sides of a triangle be represented by the vectors $\hat{i}+2 \hat{j}+2 \hat{k}$ and $3 \hat{i}-2 \hat{j}+\hat{k}$, then prove that the area of the triangle is $\frac{5}{2} \sqrt{5}$ square units.
View full solution →Find a vector perpendicular to vectors $\overrightarrow{ a }=\hat{ i }+2 \hat{ j }-3 \hat{ k }$ and $\overrightarrow{ b }=3 \hat{ i }-\hat{ j }+2 \hat{ k }$, whose magnitude is $\sqrt{171}$.
View full solution →Find the area of a triangle whose vertices are $A(1,1,1), B(1,2,3)$ and $C(2,3,3)$.
View full solution →Let $\overrightarrow{ a }=\hat{ i }+4 \hat{ j }+2 \hat{ k }, \overrightarrow{ b }=3 \hat{ i }-2 \hat{ j }+7 \hat{ k }$ and $\overrightarrow{ c }= 2 \hat{ i }-\hat{ j }+4 \hat{ k }$.Find a vector $\vec{d}$ such that $i t$ is perpendicular to both $\vec{a}$ and $\vec{b}$ and $\vec{c} \cdot \vec{d}=27$.
View full solution →Vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are such that $\vec{a}+\vec{b}+\vec{c}=0$ and $|\vec{a}|=3,|\vec{b}|=5$ and $|\vec{c}|=7$. Then find the angle between $\vec{a}$ and $\vec{b}$.
View full solution →For any vector $\vec{a}$, prove that :
$
|\overrightarrow{ a } \times \hat{ i }|^2+|\overrightarrow{ a } \times \hat{ j }|^2+|\overrightarrow{ a } \times \hat{ k }|^2= 2 |\overrightarrow{ a }|^2
$
View full solution →A vector whose initial and terminal points coincide, is called __________ .
If $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}$ and $\vec{b}=3 \hat{i}+5 \hat{j}-2 \hat{k}$, then the value of $|\vec{a} \times \vec{b}|$ will be ___________ .
View full solution →The area of triangle ABC will be __________ .
If $\vec{a}$ and $\vec{b}$ are two non-zero vactors and $\vec{a} \times \vec{b}=0$ if and only if $\vec{a}$ and $\vec{b}$ are. ___________ .
View full solution →If $\vec{a}$ is a unit vector and $(\vec{x}-\vec{a}) \cdot(\vec{x}+\vec{a})=15$, then the value of $|\vec{x}|$ will be __________ .
View full solution →