Vector Algebra · Mathematics · NDA
MCQ (Single Correct Answer)
1
The area of the square, one of whose diagonals is $$3\widehat i + 4\widehat j$$, is
NDA 2015 Paper 2
2
ABCD is a parallelogram and P is the point of intersection of the diagonals. If O is the origin, then OA + OB + OC + OD is equal to
NDA 2015 Paper 2
3
If b and c are the position vectors of the points B and C respectively, then the position vector of the point D such that BD = 4 BC is
NDA 2015 Paper 2
4
If the position vector a of the point (5, n) is such that | a | = 13, then the value(s) of n can be
NDA 2015 Paper 2
5
If | a | = 2 and | b | = 3, then, | a $$\times$$ b |2 + | a . b |2 is equal to
NDA 2015 Paper 2
6
Consider the following inequalities in respect of vectors a and b
1. | a + b | $$\le$$ | a | + | b |
2. | a $$-$$ b | $$\ge$$ | a | $$-$$ | b |
Which of the above is/are correct?
1. | a + b | $$\le$$ | a | + | b |
2. | a $$-$$ b | $$\ge$$ | a | $$-$$ | b |
Which of the above is/are correct?
NDA 2015 Paper 2
7
If the magnitude of difference of two unit vectors is $$\sqrt 3 $$, then the magnitude of sum of the two vectors is
NDA 2015 Paper 2
8
If the vectors $$\alpha \widehat i + \alpha \widehat j + \gamma \widehat k$$, $$\widehat j + \widehat k$$ and $$\gamma \widehat i + \gamma \widehat j + \beta \widehat k$$ lie on a plane, where $$\alpha$$, $$\beta$$ and $$\gamma$$ are distinct non-negative numbers, then $$\gamma$$ is
NDA 2015 Paper 2
9
The vectors a, b, c and d are such that a $$\times$$ b = c $$\times$$ d and a $$\times$$ c = b $$\times$$ d. Which of the following is/are correct?
1. (a $$-$$ d) $$\times$$ (b $$-$$ c) = 0
2. (a $$\times$$ b) $$\times$$ (c $$\times$$ d) = 0
Select the correct answer using the code given below.
1. (a $$-$$ d) $$\times$$ (b $$-$$ c) = 0
2. (a $$\times$$ b) $$\times$$ (c $$\times$$ d) = 0
Select the correct answer using the code given below.
NDA 2015 Paper 2
10
What is c equal to ?
NDA 2016 Paper 2
11
If $$d = x\widehat i + y\widehat j + z\widehat k$$, then which of the following equations is/are correct?
I. $$y - x = 4$$
II. $$2z - 3 = 0$$
Select the correct answer using the code given below.
I. $$y - x = 4$$
II. $$2z - 3 = 0$$
Select the correct answer using the code given below.
NDA 2016 Paper 2
12
What is a . b + b . c + c . a equal to?
NDA 2016 Paper 2
13
What is the angle between a and b?
NDA 2016 Paper 2
14
In a right-angled triangle ABC, if the hypotenuse AB = p, then what is AB . AC + BC . BA + CA . CB equal to ?
NDA 2016 Paper 2
15
A force $$F = 3\widehat i + 2\widehat j - 4\widehat k$$ is applied at the point (1, $$-$$ 1, 2). What is the moment of the force about the point (2, $$-$$1, 3) ?
NDA 2016 Paper 2
16
What is a vector of unit length orthogonal to both the vectors $$\widehat i + \widehat j + \widehat k$$ and $$2\widehat i + 3\widehat j - \widehat k$$ ?
NDA 2016 Paper 1
17
If a, b and c are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then which one of the following is correct?
NDA 2016 Paper 1
18
What is the area of the parallelogram having diagonals $$3\widehat i + \widehat j - 2\widehat k$$ and $$\widehat i - 3\widehat j + 4\widehat k$$ ?
NDA 2016 Paper 1
19
What is $$\cos \left( {{\theta \over 2}} \right)$$ equal to ?
NDA 2016 Paper 1
20
What is $$\sin \left( {{\theta \over 2}} \right)$$ equal to ?
NDA 2016 Paper 1
21
Let $$\alpha = \widehat i + 2\widehat j - \widehat k$$, $$\beta = 2\widehat i - \widehat j + 3\widehat k$$ and $$\gamma = 2\widehat i + \widehat j + 6\widehat k$$ be three vectors. If $$\alpha$$ and $$\beta$$ are both perpendicular to the vector $$\delta$$ and $$\delta \,.\,\gamma = 10$$, then what is the magnitude of $$\delta$$ ?
NDA 2017 Paper 2
22
If $$\widehat a$$ and $$\widehat b$$ are two unit vectors then the vector ($$\widehat a$$ + $$\widehat b$$) $$\times$$ ($$\widehat a$$ $$\times$$ $$\widehat b$$) is parallel to
NDA 2017 Paper 2
23
A force $$F = \widehat i + 3\widehat j + 2\widehat k$$ acts on a particle to displace it from the point $$A(\widehat i + 2\widehat j - 3\widehat k)$$ to the point $$B(3\widehat i - \widehat j + 5\widehat k)$$. The work done by the force will be
NDA 2017 Paper 2
24
For any vector a
$${\left| {a \times \widehat i} \right|^2} + {\left| {a \times \widehat j} \right|^2} + {\left| {a \times \widehat k} \right|^2}$$ is equal to ?
$${\left| {a \times \widehat i} \right|^2} + {\left| {a \times \widehat j} \right|^2} + {\left| {a \times \widehat k} \right|^2}$$ is equal to ?
NDA 2017 Paper 2
25
If the vectors $$a\widehat i + \widehat j + \widehat k$$, $$\widehat i + b\widehat j + \widehat k$$ and $$\widehat i + \widehat j + c\widehat k$$ (a, b, c $$\ne$$ 1) are coplanar, then the value of $${1 \over {1 - a}} + {1 \over {1 - b}} + {1 \over {1 - c}}$$ is equal to
NDA 2017 Paper 2
26
If $$a = \widehat i - \widehat j + \widehat k$$, $$b = 2\widehat i + 3\widehat j + 2\widehat k$$ and $$c = \widehat i + m\widehat j + n\widehat k$$ are three coplanar vectors and $$\left| c \right| = \sqrt 6 $$, then which one of the following is correct?
NDA 2017 Paper 1
27
Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin. What is OA + OB + OC + OD equal to ?
NDA 2017 Paper 1
28
ABCD is a quadrilateral whose diagonals are AC and BD. Which one of the following is correct?
NDA 2017 Paper 1
29
If a $$\times$$ b = c and b $$\times$$ c = a, then which one of the following is correct
NDA 2017 Paper 1
30
If $$a = 2\widehat i + 3\widehat j + 4\widehat k$$ and $$b = 3\widehat i + 2\widehat j - \lambda \widehat k$$ are perpendicular, then what is the value of $$\lambda$$ ?
NDA 2017 Paper 1
31
Let | a | $$\ne$$ 0, | b | $$\ne$$ 0. (a + b) . (a + b) = | a |2 + | b |2 holds if and only if
NDA 2018 Paper 2
32
If $$r = x\widehat i + y\widehat j + z\widehat k$$, then what is $$r\,.\,\left( {\widehat i + \widehat j + \widehat k} \right)$$ equal to ?
NDA 2018 Paper 2
33
A unit vector perpendicular to each of the vectors $$2\widehat i - \widehat j + \widehat k$$ and $$3\widehat i - 4\widehat j - \widehat k$$ is
NDA 2018 Paper 2
34
If | a | = 3, | b | = 4 and | a $$-$$ b | = 5, then what is the value of | a + b | ?
NDA 2018 Paper 2
35
Let a, b and c be three mutually perpendicular vectors each of unit magnitude. If
A = a + b + c, b = a $$-$$ b + c and C = a $$-$$ b $$-$$ c, then which one of the following is correct?
A = a + b + c, b = a $$-$$ b + c and C = a $$-$$ b $$-$$ c, then which one of the following is correct?
NDA 2018 Paper 2
36
What is (a $$-$$ b) $$\times$$ (a + b) equal to ?
NDA 2018 Paper 2
37
A spacecraft located at $$\widehat i + 2\widehat j + 3\widehat k$$ is subjected to a force $$\lambda \widehat k$$ by firing a rocket. The spacecraft is subjected to a moment of magnitude
NDA 2018 Paper 2
38
In a triangle ABC, if taken in order, consider the following statements
1. AB + BC + CA = 0
2. AB + BC $$-$$ CA = 0
3. AB $$-$$ BC + CA = 0
4. BA $$-$$ BC + CA = 0
How many of the above statement are correct?
1. AB + BC + CA = 0
2. AB + BC $$-$$ CA = 0
3. AB $$-$$ BC + CA = 0
4. BA $$-$$ BC + CA = 0
How many of the above statement are correct?
NDA 2018 Paper 2
39
If a and b are vectors such that | a | = 2, | b | = 7 and a $$\times$$ b = $$3\widehat i + 2\widehat j + 6\widehat k$$, then what is the acute angle between a and b?
NDA 2018 Paper 1
40
Let p and q be the position vectors of the points P and Q respectively with respect to origin O. The points R and S divide PQ internally and externally respectively in the ratio 2 : 3. If OR and OS are perpendicular, then which one of the following is correct?
NDA 2018 Paper 1
41
What is the moment about the point $$\widehat i + 2\widehat j - \widehat k$$ of a force represented by $$3\widehat i + \widehat k$$ acting through the point $$2\widehat i - \widehat j + 3\widehat k$$ ?
NDA 2018 Paper 1
42
If a + 2b + 3c = 0 and a $$\times$$ b + b $$\times$$ c + c $$\times$$ a = $$\lambda$$ (b $$\times$$ c), then what is the value of $$\lambda$$ ?
NDA 2018 Paper 1
43
If the vectors K and A are parallel to each other, then what is kK $$\times$$ A equal to ?
NDA 2018 Paper 1
44
Consider the following statements.
1. The magnitude of a $$\times$$ b is same as the area of a triangle with sides a and b.
2. if a $$\times$$ b = 0, where a $$\ne$$ 0, b $$\ne$$ 0, then a = $$\lambda$$ b.
Which of the above statements is/are correct?
1. The magnitude of a $$\times$$ b is same as the area of a triangle with sides a and b.
2. if a $$\times$$ b = 0, where a $$\ne$$ 0, b $$\ne$$ 0, then a = $$\lambda$$ b.
Which of the above statements is/are correct?
NDA 2019 Paper 2
45
If a and b are unit vectors and $$\theta$$ is the angle between them, then what is $${\sin ^2}\left( {{\theta \over 2}} \right)$$ equal to ?
NDA 2019 Paper 2
46
Consider the following equations for two vectors a and b.
1. $$(a + b).(a - b) = {\left| a \right|^2} - {\left| b \right|^2}$$
2. $$(\left| {a + b} \right|)(\left| {a - b} \right|) = {\left| a \right|^2} - {\left| b \right|^2}$$
3. $$\left| {a\,.\,b} \right| + \left| {a \times b} \right| = {\left| a \right|^2}{\left| b \right|^2}$$
Which of the above statements are correct?
1. $$(a + b).(a - b) = {\left| a \right|^2} - {\left| b \right|^2}$$
2. $$(\left| {a + b} \right|)(\left| {a - b} \right|) = {\left| a \right|^2} - {\left| b \right|^2}$$
3. $$\left| {a\,.\,b} \right| + \left| {a \times b} \right| = {\left| a \right|^2}{\left| b \right|^2}$$
Which of the above statements are correct?
NDA 2019 Paper 2
47
If the magnitude of the sum of two non-zero vectors is equal to magnitude of their difference, then which one of the following is correct?
NDA 2019 Paper 2
48
What is the scalar projection of
$$a = \widehat i - 2\widehat j + \widehat k$$ on
$$b = 4\widehat i - 4\widehat j + 7\widehat k$$ ?
$$a = \widehat i - 2\widehat j + \widehat k$$ on
$$b = 4\widehat i - 4\widehat j + 7\widehat k$$ ?
NDA 2019 Paper 2
49
If $$a = \widehat i - 2\widehat j + 5\widehat k$$ and $$b = 2\widehat i + \widehat j - 3\widehat k$$, then what is $$(b - a).(3a + b)$$ equal to ?
NDA 2019 Paper 1
50
If the position vectors of points A and B are $$3\widehat i - 2\widehat j + \widehat k$$ and $$2\widehat i + 4\widehat j - 3\widehat k$$ respectively, then what is the length of AB ?
NDA 2019 Paper 1
51
If in a right angled triangle ABC, hypotenuse AC = p, then what is AB . AC + BC . BA + CA . CB equal to?
NDA 2019 Paper 1
52
The sine of the angle between vectors $$a = 2\widehat i - 6\widehat j - 3\widehat k$$ and $$b = 4\widehat i + 3\widehat j - \widehat k$$ is
NDA 2019 Paper 1
53
What is the value of $$\lambda$$ for which the vectors $$3\widehat i + 4\widehat j - \widehat k$$ and $$ - 2\widehat i + \lambda \widehat j + 10\widehat k$$ are perpendicular?
NDA 2019 Paper 1
54
If the direction cosines <l, m, n> of a line are connected by relation $l + 2m + n = 0, 2l - 2m + 3n = 0$, then what is the value of $l^{2} + m^{2} - n^{2}$?
NDA Mathematics 21 April 2024
55
Let $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} - \hat{k}$. If $\vec{a} \times (\vec{b} \times \vec{a}) = \alpha \hat{i} - \beta \hat{j} + \gamma \hat{k}$, then what is the value of $\alpha + \beta + \gamma$?
NDA Mathematics 21 April 2024
56
If a vector of magnitude 2 units makes an angle $\frac{\pi}{3}$ with $2\hat{i}$, $\frac{\pi}{4}$ with $3\hat{j}$ and an acute angle $\theta$ with $4\hat{k}$, then what are the components of the vector?
NDA Mathematics 21 April 2024
57
Consider the following in respect of moment of a force:
1. The moment of force about a point is independent of point of application of force.
2. The moment of a force about a line is a vector quantity.
Which of the statements given above is/are correct?
NDA Mathematics 21 April 2024
58
For any vector $\vec{r}$, what is $\left(\vec{r}\cdot\hat{i}\right)\left(\vec{r}\times\hat{i}\right) + \left(\vec{r}\cdot\hat{j}\right)\left(\vec{r}\times\hat{j}\right) + \left(\vec{r}\cdot\hat{k}\right)\left(\vec{r}\times\hat{k}\right)$ equal to?
NDA Mathematics 21 April 2024
59
Let $\vec{a}$ and $\vec{b}$ be two vectors of magnitude 4 inclined at an angle $\frac{\pi}{3}$, then what is the angle between $\vec{a}$ and $\vec{a} - \vec{b}$?
NDA Mathematics 21 April 2024
60
What is the length of projection of the vector $\rm \hat{i}+2 \hat{j}+3 \hat{k}$ on the vector $\rm2 \hat{i}+3 \hat{j}-2 \hat{k}$ ?
NDA Mathematics 3 September 2023
61
If $\rm (\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=144$ and $\rm|\vec{b}|=4 $, then what is the value of $\rm|\vec{a}|$ ?
NDA Mathematics 3 September 2023
62
If θ is the angle between vectors $\vec{a}$ and $ \vec{b}$ such that $\vec{a} \cdot \vec{b} \geq 0$, then which one of the following is correct?
NDA Mathematics 3 September 2023
63
The vectors $\rm 60 \hat{i}+3 \hat{j}, 40 \hat{i}-8 \hat{j}$ and $\rm \beta \hat{i}-52 \hat{j}$ are collinear if :
NDA Mathematics 3 September 2023
64
Consider the following in respect of the vectors $\rm \vec{a}=(0,1,1)$ and $\rm \vec{b}=(1,0,1) $ :
1. The number of unit vectors perpendicular to both $\rm \vec{a}$ and $\rm \vec{b}$ is only one.
2. The angle between the vectors is $\frac{\pi}{3}$.
Which of the statements given above is/are correct?
NDA Mathematics 3 September 2023
65
What is $\vec{b}$ equal to ?
NDA Mathematics 16 April 2023
66
What is the angle between $(\vec{a}+\vec{b})$ and $\vec{c}$ ?
NDA Mathematics 16 April 2023
67
What is cosα equal to ?
NDA Mathematics 16 April 2023
68
What is cos2β + cos2γ equal to ?
NDA Mathematics 16 April 2023
69
Consider the following points :
1. (-1, -3, 1)
2. (-1, 3, 2)
3. (-2, 5, 3)
Which of the above points lie on the line joining A and B ?
NDA Mathematics 16 April 2023
70
What is the magnitude of $\overrightarrow{A B}$ ?
NDA Mathematics 16 April 2023
71
PQRS is a parallelogram. If $\overrightarrow{\text{PR}}=\vec{\text{a}}$ and $\overrightarrow{\text{QS}}=\vec{\text{b}}$, then what is $\overrightarrow{\text{PQ}}$ equal to ?
NDA Mathematics 4 September 2022
72
Let $\vec{\text{a}}$ and $\vec{\text{b}}$ are two unit vectors such that $\vec{\text{a}}+2 \vec{\text{b}}$ and $5\vec{\text{a}}−4\vec{\text{b}}$ are perpendicular. What is the angle between $\vec{\text{a}}$ and $\vec{\text{b}}$ ?
NDA Mathematics 4 September 2022
73
Let $\vec{\text{a}}$, $\vec{\text{b}}$ and $\vec{\text{c}}$ be unit vectors lying on the same plane. What is $\{(3\vec{\text{a}} + 2\vec{\text{b}}) × (5\vec{\text{a}} − 4\vec{\text{c}})\}⋅(\vec{\text{b}} + 2\vec{\text{c}})$ equal to ?
NDA Mathematics 4 September 2022
74
What are the values of x for which the angle between the vectors 2x2$\hat{\text{i}}$ + 3x$\hat{\text{j}}$ + $\hat{\text{k}}$ and $\hat{\text{i}}$ −2$\hat{\text{j}}$ + x2$\hat{\text{k}}$ is obtuse ?
NDA Mathematics 4 September 2022
75
The position vectors of vertices A, B and C of triangle ABC are respectively $\hat{\text{j}}+\hat{\text{k}}, 3\hat{\text{i}}+\hat{\text{j}+5\hat{\text{k}}}$ and $3\hat{\text{j}}+3\hat{\text{k}}$. What is angle C equal to?
NDA Mathematics 4 September 2022
76
If $4\hat i + \hat j - 3\hat k$ and $p\hat i + q\hat j - 2\hat k$ are collinear vectors, then what are the possible values of p and q respectively?
NDA Mathematics 10 April 2022
77
If $\vec a, \vec b, \vec c$, are the position vectors of the vertices A, B, C respectively of a triangle ABC and G is the centroid of the triangle, then what is $\overrightarrow{AG}$ equal to ? .
NDA Mathematics 10 April 2022
78
Consider the following statements :
1. Dot product over vector addition is distributive
2. Cross product over vector addition is distributive
3. Cross product of vectors is associative
Which of the above statements is/are correct ?
NDA Mathematics 10 April 2022
79
Let $\vec{a}, \vec{b}, \vec{c}$ be three non-zero vectors such that $\vec{a}\times \vec{b} = \vec{c} $. Consider the following statements:
1. $\vec a$ is unique if $\vec b$ and $\vec c$ are given
2. $\vec c$ is unique if $\vec a$ and $\vec b$ are given
Which of the above statements is/are correct?
NDA Mathematics 10 April 2022
80
Let $\vec a$ and $\vec b$ be two unit vectors such that $|\vec a - \vec b|<2.$ If 2θ is the angle between $\vec a$ and $\vec b,$ then which one of the following is correct ?
NDA Mathematics 10 April 2022
81
What is the projection of the line segment joining A(1, 7, -5) and B(-3, 4, -2) on y-axis?
NDA Mathematics 18 April 2021
82
A vector $\vec r=a \hat i+b \hat j$ is equally inclined to both x and y axes. If the magnitude of the vector is 2 units, then what are the values of a and b respectively?
NDA Mathematics 18 April 2021
83
Consider the following statements in respect of a vector $\vec c=\vec a+\vec b$, where $|\vec a|=|\vec b|\ne0$:
1. $\vec c$ is perpendicular to $(\vec a-\vec b).$
2. $\vec c$ is perpendicular to $\vec a \times \vec b.$
Which of the above statement is/are correct?
NDA Mathematics 18 April 2021
84
If $\vec a \:and\: \vec b$ are two vectors such that $|\vec a + \vec b|= |\vec a - \vec b|=4,$ then which one of the following is correct?
NDA Mathematics 18 April 2021
85
If $\vec a, \vec b\:and \: \vec c$ are coplanar, then what is $(2\vec a\times 3\vec b)\cdot4\vec c+(5\vec b\times 3\vec c)\cdot6\vec a$ equal to?
NDA Mathematics 18 April 2021
86
Consider the following statements:
1. The cross product of two unit vectors is always a unit vector.
2. The dot product of two unit vectors is always unity.
3. The magnitude of sum of two unit vectors is always greater than the magnitude of their difference.
Which of the above statements are not correct?
NDA Mathematics 18 April 2021