Vectors · NDA previous year questions

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 are the direction cosines of $\bar{r}$?

1. A.$⟨\frac{1}{2},\frac{\sqrt{3}}{2},0⟩$
2. B.$⟨\frac{1}{2},-\frac{\sqrt{3}}{2},0⟩$
3. C.$⟨\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0⟩$
4. D.$⟨-\frac{1}{2},\frac{\sqrt{3}}{2},0⟩$

If the angle between the vectors $\hat{i}-m\hat{j}$ and $\hat{j}+\hat{k}$ is $\frac{\pi }{3}$, then what is the value of $m$?

1. A.0
2. B.2
3. C.$-2$
4. D.None of the above

What is the vector perpendicular to both the vectors $\hat{i}-\hat{j}$ and $\hat{i}$?

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

The position vectors of the points $A$ and $B$ are respectively $3\hat{i}-5\hat{j}+2\hat{k}$ and $\hat{i}+\hat{j}-\hat{k}$. What is the length of $AB$?

1. A.11
2. B.9
3. C.7
4. D.6

The vectors $\hat{i}-2x\hat{j}-3y\hat{k}$ and $\hat{i}+3x\hat{j}+2y\hat{k}$ are orthogonal to each other. Then the locus of the point $(x,y)$ is

1. A.hyperbola
2. B.ellipse
3. C.parabola
4. D.circle

What is the value of $p$ for which the vector $p(2\hat{i}-\hat{j}+2\hat{k})$ is of 3 units length?

1. A.1
2. B.2
3. C.3
4. D.6

If $\vec{a}=2\hat{i}+2\hat{j}+3\hat{k},\vec{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\vec{c}=3\hat{i}+\hat{j}$ are three vectors such that $\vec{a}+t\vec{b}$ is perpendicular to $\vec{c}$, then what is $t$ equal to?

1. A.8
2. B.6
3. C.4
4. D.2

For any vector $\vec{a}$, what is $(\vec{a}\cdot \hat{i})\hat{i}+(\vec{a}\cdot \hat{j})\hat{j}+(\vec{a}\cdot \hat{k})\hat{k}$ equal to ?

1. A.$\vec{a}$
2. B.$3\vec{a}$
3. C.$-\vec{a}$
4. D.$\vec{0}$

If the magnitude of $\vec{a}\times \vec{b}$ equals to $\vec{a}\cdot \vec{b}$, then which one of the following is correct ?

1. A.$\vec{a}=\vec{b}$
2. B.The angle between $\vec{a}$ and $\vec{b}$ is $45^{\circ }$
3. C.$\vec{a}$ is parallel to $\vec{b}$
4. D.$\vec{a}$ is perpendicular to $\vec{b}$

If $|\vec{a}|=\sqrt{2},|\vec{b}|=\sqrt{3}$ and $|\vec{a}+\vec{b}|=\sqrt{6}$, then what is $|\vec{a}-\vec{b}|$ equal to ?

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

Which one of the following vectors is normal to the vector $\hat{i}+\hat{j}+\hat{k}$ ?

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

If $\theta$ is the angle between the vectors $4(\hat{i}-\hat{k})$ and $\hat{i}+\hat{j}+\hat{k}$, then what is $(\sin \theta +\cos \theta )$ equal to ?

1. A.0
2. B.$\frac{1}{2}$
3. C.1
4. D.2

The average marks obtained by the students in a class are 43. If the average marks obtained by 25 boys are 40 and the average marks obtained by the girl students are 48, then what is the number of girl students in the class ?

1. A.15
2. B.17
3. C.18
4. D.20

EFGH is a rhombus such that the angle EFG is $60^{\circ }$. The magnitude of vectors $\overrightarrow{FH}$ and $\{m\overrightarrow{EG}\}$ are equal where m is a scalar. What is the value of m?

1. A.3
2. B.1.5
3. C.$\sqrt{2}$
4. D.$\sqrt{3}$

If $\vec{a}\cdot \vec{b}=0$ and $\vec{a}\times \vec{b}=\vec{0}$ then which one of the following is correct?

1. A.$\vec{a}$ is parallel to $\vec{b}$
2. B.$\vec{a}$ is perpendicular to $\vec{b}$
3. C.$\vec{a}=\vec{0}$ or $\vec{b}=\vec{0}$
4. D.None of the above

The vector $\vec{a}\times (\vec{b}\times \vec{a})$ is coplanar with:

1. A.$\vec{a}$ only
2. B.$\vec{b}$ only
3. C.Both $\vec{a}$ and $\vec{b}$
4. D.Neither $\vec{a}$ nor $\vec{b}$

Consider the following: 1. $4\hat{i}\times 3\hat{i}=\vec{0}$ 2. $\frac{4\hat{i}}{3\hat{i}}=\frac{4}{3}$ 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

What is the value of $\lambda$ for which $(\lambda \hat{i}+\hat{j}-\hat{k})\times (3\hat{i}-2\hat{j}+4\hat{k})=(2\hat{i}-11\hat{j}-7\hat{k})$?

1. A.2
2. B.$-2$
3. C.1
4. D.7

The magnitude of the scalar p for which the vector $p(-3\hat{i}-2\hat{j}+13\hat{k})$ is of unit length is:

1. A.1/8
2. B.1/64
3. C.$\sqrt{182}$
4. D.$1/\sqrt{182}$

The vector $2\hat{j}-\hat{k}$ lies:

1. A.in the plane of XY
2. B.in the plane of YZ
3. C.in the plane of XZ
4. D.along the X-axis

ABCD is a parallelogram. If $\overrightarrow{AB}=\vec{a}$, $\overrightarrow{BC}=\vec{b}$, then what is $\overrightarrow{BD}$ equal to?

1. A.$\vec{a}+\vec{b}$
2. B.$\vec{a}-\vec{b}$
3. C.$-\vec{a}-\vec{b}$
4. D.$-\vec{a}+\vec{b}$

If $\vec{a}=(2,1,-1),\vec{b}=(1,-1,0),\vec{c}=(5,-1,1)$, then what is the unit vector parallel to $\vec{a}+\vec{b}-\vec{c}$ in the opposite direction ?

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

If the magnitudes of two vectors $\vec{a}$ and $\vec{b}$ are equal then which one of the following is correct ?

1. A.$(\vec{a}+\vec{b})$ is parallel to $(\vec{a}-\vec{b})$
2. B.$(\vec{a}+\vec{b})\cdot (\vec{a}-\vec{b})=1$
3. C.$(\vec{a}+\vec{b})$ is perpendicular to $(\vec{a}-\vec{b})$
4. D.None of the above

What is the value of m if the vectors $2\hat{i}-\hat{j}+\hat{k},\hat{i}+2\hat{j}-3\hat{k}$ and $3\hat{i}+m\hat{j}+5\hat{k}$ are coplanar ?

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

If $|\vec{a}|=10,|\vec{b}|=2$ and $\vec{a}\bullet \vec{b}=12$, then what is the value of $|\vec{a}\times \vec{b}|$?

1. A.12
2. B.16
3. C.20
4. D.24