Vectors · NDA previous year questions

If $|\vec{a}|=2$ and $|\vec{b}|=3$, then $|\vec{a}\times \vec{b}{|}^2+|\vec{a}\cdot \vec{b}{|}^2$ is equal to

1. A.72
2. B.64
3. C.48
4. D.36

Consider the following inequalities in respect of vectors $\vec{a}$ and $\vec{b}$: 1. $|\vec{a}+\vec{b}|\le |\vec{a}|+|\vec{b}|$ 2. $|\vec{a}-\vec{b}|\ge |\vec{a}|-|\vec{b}|$ Which of the above is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

If the magnitude of difference of two unit vectors is $\sqrt{3}$, then the magnitude of sum of the two vectors is

1. A.$\frac{1}{2}$ unit
2. B.1 unit
3. C.2 unit
4. D.3 unit

If the vectors $\alpha \hat{i}+\alpha \hat{j}+\gamma \hat{k},\hat{i}+\hat{k}$ and $\gamma \hat{i}+\gamma \hat{j}+\beta \hat{k}$ lie on a plane, where $\alpha ,\beta$ and $\gamma$ are distinct non-negative numbers, then $\gamma$ is

1. A.Arithmetic mean of $\alpha$ and $\beta$
2. B.Geometric mean of $\alpha$ and $\beta$
3. C.Harmonic mean of $\alpha$ and $\beta$
4. D.None of the above

The vectors $\vec{a},\vec{b},\vec{c}$ and $\vec{d}$ are such that $\vec{a}\times \vec{b}=\vec{c}\times \vec{d}$ and $\vec{a}\times \vec{c}=\vec{b}\times \vec{d}$. Which of the following is/are correct? 1. $(\vec{a}-\vec{d})\times (\vec{b}-\vec{c})=\vec{0}$ 2. $(\vec{a}\times \vec{b})\times (\vec{c}\times \vec{d})=\vec{0}$ Select the correct answer using the code given below:

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

The adjacent sides $AB$ and $AC$ of a triangle $ABC$ are represented by the vectors $-2\hat{i}+3\hat{j}+2\hat{k}$ and $-4\hat{i}+5\hat{j}+2\hat{k}$ respectively. The area of the triangle $ABC$ is

1. A.6 square units
2. B.5 square units
3. C.4 square units
4. D.3 square units

A force $\vec{F}=3\hat{i}+4\hat{j}-3\hat{k}$ is applied at the point $P$, whose position vector is $\vec{r}=2\hat{i}-2\hat{j}-3\hat{k}$. What is the magnitude of the moment of the force about the origin?

1. A.23 units
2. B.19 units
3. C.18 units
4. D.21 units

Given that the vectors $\vec{\alpha }$ and $\vec{\beta }$ are non-collinear. The values of $x$ and $y$ for which $\vec{u}-\vec{v}=\vec{w}$ holds true if $\vec{u}=2x\vec{\alpha }+y\vec{\beta }$, $\vec{v}=2y\vec{\alpha }+3x\vec{\beta }$ and $\vec{w}=2\vec{\alpha }-5\vec{\beta }$, are

1. A.$x=2,y=1$
2. B.$x=1,y=2$
3. C.$x=-2,y=1$
4. D.$x=-2,y=-1$

If $|\vec{a}|=7$, $|\vec{b}|=11$ and $|\vec{a}+\vec{b}|=10\sqrt{3}$, then $|\vec{a}-\vec{b}|$ is equal to

1. A.40
2. B.10
3. C.$4\sqrt{10}$
4. D.$2\sqrt{10}$

Let $\alpha ,\beta ,\gamma$ be distinct real numbers. The points with position vectors $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$, $\beta \hat{i}+\gamma \hat{j}+\alpha \hat{k}$ and $\gamma \hat{i}+\alpha \hat{j}+\beta \hat{k}$

1. A.are collinear
2. B.form an equilateral triangle
3. C.form a scalene triangle
4. D.form a right-angled triangle

If $\vec{a}+\vec{b}+\vec{c}=\vec{0}$, then which of the following is/are correct? 1. $\vec{a},\vec{b},\vec{c}$ are coplanar. 2. $\vec{a}\times \vec{b}=\vec{b}\times \vec{c}=\vec{c}\times \vec{a}$ Select the correct answer using the code given below.

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

If $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$, then which one of the following is correct?

1. A.$\vec{a}=\lambda \vec{b}$ for some scalar $\lambda$
2. B.$\vec{a}$ is parallel to $\vec{b}$
3. C.$\vec{a}$ is perpendicular to $\vec{b}$
4. D.$\vec{a}=\vec{b}=\vec{0}$

If $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$, then which one of the following is correct?

1. A.$|\vec{a}|=|\vec{b}|$.
2. B.$\vec{a}$ is parallel to $\vec{b}$.
3. C.$\vec{a}$ is perpendicular to $\vec{b}$.
4. D.$\vec{a}$ is a unit vector.

Which one of the following is the unit vector perpendicular to both $\vec{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$?

1. A.$\frac{\hat{i}+\hat{j}}{\sqrt{2}}$
2. B.$\hat{k}$
3. C.$\frac{\hat{j}+\hat{k}}{\sqrt{2}}$
4. D.$\frac{\hat{i}-\hat{j}}{\sqrt{2}}$

What is the interior acute angle of the parallelogram whose sides are represented by the vectors $\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\hat{k}$ and $\frac{1}{\sqrt{2}}\hat{i}-\frac{1}{\sqrt{2}}\hat{j}+\hat{k}$?

1. A.$60^{\circ }$
2. B.$45^{\circ }$
3. C.$30^{\circ }$
4. D.$15^{\circ }$

For what value of $\lambda$ are the vectors $\lambda \hat{i}+(1+\lambda )\hat{j}+(1+2\lambda )\hat{k}$ and $(1-\lambda )\hat{i}+\lambda \hat{j}+2\hat{k}$ perpendicular?

1. A.-1/3
2. B.1/3
3. C.2/3
4. D.1

For the next four (04) items that follow: $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ such that $|\vec{a}|=3$, $|\vec{b}|=5$ and $|\vec{c}|=7$. What is the angle between $\vec{a}$ and $\vec{b}$?

1. A.$\pi /6$
2. B.$\pi /4$
3. C.$\pi /3$
4. D.$\pi /2$

For the next four (04) items that follow: $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ such that $|\vec{a}|=3$, $|\vec{b}|=5$ and $|\vec{c}|=7$. What is $\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}$ equal to?

1. A.-83
2. B.-83/2
3. C.75
4. D.-75/2

For the next four (04) items that follow: $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ such that $|\vec{a}|=3$, $|\vec{b}|=5$ and $|\vec{c}|=7$. What is cosine of the angle between $\vec{b}$ and $\vec{c}$?

1. A.11/12
2. B.13/14
3. C.-11/12
4. D.-13/14

For the next four (04) items that follow: $\vec{a}+\vec{b}+\vec{c}=\vec{0}$ such that $|\vec{a}|=3$, $|\vec{b}|=5$ and $|\vec{c}|=7$. What is $|\vec{a}+\vec{b}|$ equal to?

1. A.7
2. B.8
3. C.10
4. D.11

For the next two (02) items that follow: Consider the vectors $\vec{a}=\hat{i}-2\hat{j}+\hat{k}$ and $\vec{b}=4\hat{i}-4\hat{j}+7\hat{k}$. What is the scalar projection of $\vec{a}$ on $\vec{b}$?

1. A.1
2. B.19/9
3. C.17/9
4. D.23/9

For the next two (02) items that follow: Consider the vectors $\vec{a}=\hat{i}-2\hat{j}+\hat{k}$ and $\vec{b}=4\hat{i}-4\hat{j}+7\hat{k}$. What is the vector perpendicular to both the vectors?

1. A.$-10\hat{i}-3\hat{j}+4\hat{k}$
2. B.$-10\hat{i}+3\hat{j}+4\hat{k}$
3. C.$10\hat{i}-3\hat{j}+4\hat{k}$
4. D.None of the above

For the next two (02) items that follow: Let $|\bar{a}|=7$, $|\bar{b}|=11$, $|\bar{a}+\bar{b}|=10\sqrt{3}$ What is $|\bar{a}-\bar{b}|$ equal to?

1. A.$2\sqrt{2}$
2. B.$2\sqrt{10}$
3. C.5
4. D.10

For the next two (02) items that follow: Let $|\bar{a}|=7$, $|\bar{b}|=11$, $|\bar{a}+\bar{b}|=10\sqrt{3}$ What is the angle between $(\bar{a}+\bar{b})$ and $(\bar{a}-\bar{b})$?

1. A.$\frac{\pi }{2}$
2. B.$\frac{\pi }{3}$
3. C.$\frac{\pi }{6}$
4. D.None of the above

For the next two (02) items that follow: Let a vector $\bar{r}$ make angles $60^{\circ }$, $30^{\circ }$ with x and y-axes respectively. What angle does $\bar{r}$ make with z-axis?

1. A.$30^{\circ }$
2. B.$60^{\circ }$
3. C.$90^{\circ }$
4. D.$120^{\circ }$