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 :
View full solution →The unit vector along the vector $\vec{a}=-2 \hat{i}+3 \hat{j}-\hat{k}$ is :
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:
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:
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:
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 vector directed from A to B and joining the points $A (1,2,2)$ and $B (2,3,1)$ is :
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 →View full solution →Evaluate the product $(3 \vec{a}-5 \vec{b}) \cdot(2 \vec{a}+7 \vec{b})$
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 →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 →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 vectors $\vec{a}$ and $\vec{b}$ prove that : $|\vec{a} \times \vec{b}|^2=|\vec{a}|^2|\vec{b}|^2-|\vec{a} \cdot \vec{b}|^2 .$
View full solution →Using vector method prove that in $\triangle ABC$, $\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta$
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 →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 →If the angle between two non-zero vectors $\vec{a}$ and $\vec{b}$ is $\theta$, then $\cos \theta=$ ___________ .
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. ___________ .
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