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Vector Algebra — NDA Mathematics PYQs

100 previous year questions from Vector Algebra with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

100 questionsMathematicsSolutions on every page
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Passage: Let a b = c and b c = a . Question: Consider the following statements : I. ( a b ) c +( b c ) a =( c a ) b II. ( a b ) ( b c ) b =1 Which of the statements given above is/

NDA 2026 Solution View → 2

Passage: Let a , b , c , a + b , b + c , a + b + c be unit vectors. Question: What is the angle between a and b ?

NDA 2026 Solution View → 3

Passage: Let a , b , c be unit vectors. Further, a is perpendicular to b ; c makes an angle 3 with both a and b ; and c =p a +q b +r( a b ) . Question: What is the value of r^2 ?

NDA 2026 Solution View → 4

Passage: Let a , b , c , a + b , b + c , a + b + c be unit vectors. Question: What is the angle between a and c ?

NDA 2026 Solution View → 5

Passage: Let a b = c and b c = a . Question: Consider the following statements : I. a , b , c are orthogonal in pairs. II. a , b , c are unit vectors. Which of the statements given

NDA 2026 Solution View → 6

Passage: Let a , b , c be unit vectors. Further, a is perpendicular to b ; c makes an angle 3 with both a and b ; and c =p a +q b +r( a b ) . Question: What is the value of (p+q) ?

NDA 2026 Solution View → 7

What is the area of the parallelogram whose sides are represented by the vectors i + 2 j + 3 k and 2 i + j + 2 k ?

NDA 2025 Solution View → 8

Let p = a - b , q = a + b . If | a | = | b | = 2 and a b = 2 , then what is the value of | p q | ?

NDA 2025 Solution View → 9

A force F = 2 i - j + 5 k is applied at the point A(1, 2, 5) . If its moment about the point B(-1, -2, 3) is 16 i - 6 j + 2 k , then what is the value of ?

NDA 2025 Solution View → 10

ABC is a triangle right-angled at B . If A(k, 1, -1) , B(2k, 0, 2) and C(2+2k, k, 1) are the vertices of the triangle, then what is the value of k ?

NDA 2025 Solution View → 11

Consider the following statements in respect of a vector d = ( a b ) c : I. d is coplanar with a and b . II. d is perpendicular to c . Which of the statements given above is/are co

NDA 2025 Solution View → 12

The position vectors of three points A , B and C are a , b and c respectively such that 3 a - 4 b + c = 0 . What is AB : BC equal to?

NDA 2025 Solution View → 13

The position vectors of three points A , B and C are a , b and c respectively, where c = ( ^2 ) a + ( ^2 ) b . What is ( a b ) + ( b c ) + ( c a ) equal to?

NDA 2025 Solution View → 14

Let a , b , ( a b ) be unit vectors. What is ( a b ) equal to?

NDA 2025 Solution View → 15

Let be the angle between two unit vectors a and b . If a +2 b is perpendicular to 5 a -4 b , then what is + 2 equal to?

NDA 2024 Solution View → 16

What is 3 +2 equal to if (2 i +6 j +27 k ) ( i + j + k ) is a null vector?

NDA 2024 Solution View → 17

Let ABCDEF be a regular hexagon. If AD =m BC and CF =n AB , then what is m n equal to?

NDA 2024 Solution View → 18

The vectors a , b and c are of the same length. If taken pairwise, they form equal angles. If a = i + j and b = j + k , then what can c be equal to? A. i + k B. - i +4 j - k 3 Sele

NDA 2024 Solution View → 19

Let a and b are two vectors of magnitude 4 inclined at an angle 3 , then what is the angle between a and a - b ?

NDA 2024 Solution View → 20

For any vector r , what is ( r i )( r i )+( r j )( r j )+( r k )( r k ) equal to?

NDA 2024 Solution View → 21

If ( a b )^2+( a b )^2=144 and | b |=4 , then what is the value of | a | ?

NDA 2023 Solution View → 22

What is the length of projection of the vector i +2 j +3 k on the vector 2 i +3 j -2 k ?

NDA 2023 Solution View → 23

If is the angle between vectors a and b such that a b 0 , then which one of the following is correct?

NDA 2023 Solution View → 24

The vectors 60 i +3 j , 40 i -8 j & i -52 j are collinear if:

NDA 2023 Solution View → 25

If | a |=7,| b |=11 and | a + b |=10 3 , then | a - b | is equal to

NDA 2015 Solution View → 26

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

NDA 2015 Solution View → 27

If | a + b |=| a - b | , then which one of the following is correct?

NDA 2015 Solution View → 28

a + b + c = 0 such that | a | 3,| b |=5 and | c |=7 . What is cosine of the angle between b and c ?

NDA 2014 Solution View → 29

a + b + c = 0 such that | a | 3,| b |=5 and | c |=7 . What is | a + b | equal to?

NDA 2014 Solution View → 30

Paragraph : Consider the vectors a = i -2 j + k and b =4 i -4 j +7 k . Question : What is the scalar projection of a on b ?

NDA 2014 Solution View → 31

Paragraph : Consider the vectors a = i -2 j + k and b =4 i -4 j +7 k . Question : What is the vector perpendicular to both the vectors?

NDA 2014 Solution View → 32

Which one of the following is the unit vector perpendicular to both a =- i + j + k and b =- i - j + k ?

NDA 2014 Solution View → 33

Paragraph : Let | a |=7, | b |=11 , | a + b |=10 3 Question : What is | a - b | equal to?

NDA 2014 Solution View → 34

Paragraph : Let | a |=7, | b |=11 , | a + b |=10 3 Question : What is the angle between ( a + b ) and ( a - b ) ?

NDA 2014 Solution View → 35

What is the interior acute angle of the parallelogram whose sides are represented by the vectors 1 2 i + 1 2 j + k and 1 2 i - 1 2 j + k ?

NDA 2014 Solution View → 36

a + b + c = 0 such that | a | 3,| b |=5 and | c |=7 . What is the angle between a and b ?

NDA 2014 Solution View → 37

For what value of are the vectors i +(1+ ) j +(1+2 ) k and (1- ) i + j +2 k perpendicular?

NDA 2014 Solution View → 38

a + b + c = 0 such that | a | 3,| b |=5 and | c |=7 . What is a b + b c + c a equal to?

NDA 2014 Solution View → 39

If | a |=2,| b |=5 and | a b |=8 , then what is a b equal to?

NDA 2014 Solution View → 40

If | a + b |=| a - b | , then which one of the following is correct?

NDA 2014 Solution View → 41

What is the area of the triangle OAB where O is the origin, OA =3 i - j + k and OB =2 i + j -3 k ?

NDA 2014 Solution View → 42

What angle does the line segment joining (5,2) and (6,-15) subtend at (0,0) ?

NDA 2013 Solution View → 43

If the angle between the vectors i -m j and j + k is 3 , then what is the value of m ?

NDA 2013 Solution View → 44

What is the vector perpendicular to both the vectors i - j and i ?

NDA 2013 Solution View → 45

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

NDA 2013 Solution View → 46

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

NDA 2013 Solution View → 47

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

NDA 2013 Solution View → 48

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

NDA 2013 Solution View → 49

What is the area of the triangle whose vertices are (0,0,0),(1,2,3) and (-3,-2,1) ?

NDA 2013 Solution View → 50

For any vector , what is ( i ) i +( j ) j +( k ) k equal to?

NDA 2013 Solution View → 51

If is perpendicular to both and where = k and =2 i +3 j +4 k , then what is equal to?

NDA 2013 Solution View → 52

If the magnitude of a b equals to a b , then which one of the following is correct?

NDA 2013 Solution View → 53

If | a |= 2 ,| b |= 3 and | a + b |= 6 , then what is | a - b | equal to?

NDA 2013 Solution View → 54

Which one of the following vectors is normal to the vector i + j + k ?

NDA 2013 Solution View → 55

If is the angle between the vectors 4( i - k ) and i + j + k , then what is ( + ) equal to?

NDA 2013 Solution View → 56

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

NDA 2012 Solution View → 57

If a b =0 and a b = 0 then which one of the following is correct?

NDA 2012 Solution View → 58

The vectors a ( b a ) is coplanar with :

NDA 2012 Solution View → 59

Consider the following : 1. 4 i 3 i = 0 2. 4 i 3 i = 4 3 Which of the above is/are correct?

NDA 2012 Solution View → 60

The vector 2 j - k lies:

NDA 2012 Solution View → 61

What is the value of for which ( i + j - k ) (3 i -2 j +4 k )=(2 i -11 j -7 k ) ?

NDA 2012 Solution View → 62

ABCD is a parallelogram. If AB = a , BC = b , then what is BD equal to?

NDA 2012 Solution View → 63

If the magnitudes of two vectors a and b are equal then which one of the following is correct?

NDA 2012 Solution View → 64

Let O be the origin and P, Q, R be the points such that PO + OQ = QO + OR . Then which one of the following is correct?

NDA 2012 Solution View → 65

If | a |=10,| b |=2 and a b =12 , then what is the value of | a b | ?

NDA 2012 Solution View → 66

If the vectors i -x j -y k and i +x j -y k are orthogonal to each other, then what is the locus of the point (x, y) ?

NDA 2012 Solution View → 67

What is the area of the rectangle having vertices A , B , C and D with position vectors - i + 1 2 j +4 k , i + 1 2 j +4 k , i - 1 2 j +4 k and - i - 1 2 j +4 k ?

NDA 2012 Solution View → 68

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

NDA 2012 Solution View → 69

If a =(2,1,-1), b =(1,-1,0), c =(5,-1,1) , then wha is the unit vector parallel to a + b - c in the opposite direction?

NDA 2012 Solution View → 70

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

NDA 2012 Solution View → 71

If a and b are two vectors such that a b =0 and a b = 0 , then which one of the following is correct?

NDA 2011 Solution View → 72

If the vectors - i -2 x j -3 y k and i -3 x j -2 y k are orthogonal to each other, then what is the locus of the point (x, y) ?

NDA 2011 Solution View → 73

If c is the unit vector perpendicular to both the vectors a and b , then what is another unit vector perpendicular to both the vectors a and b ?

NDA 2011 Solution View → 74

For what value of m are the points with position vectors 10 i +3 j , 12 i -5 j and m i +11 j collinear?

NDA 2011 Solution View → 75

For what value of m are the vectors 2 i -3 j +4 k , i +2 j - k and m i - j +2 k coplanar?

NDA 2011 Solution View → 76

What is the area of the triangle with vertices (1,2,3) , (2,5,-1) and (-1,1,2) ?

NDA 2011 Solution View → 77

The vector a lies in the plane of vectors b and c . Which one of the following is correct?

NDA 2011 Solution View → 78

If P, Q, R are three non-collinear points, then what is P Q P R equal to?

NDA 2011 Solution View → 79

If x i +y j +z k is a unit vector and x: y: z= 3 : 2: 3 , then what is the value of z ?

NDA 2011 Solution View → 80

Let p, q, r, s be respectively the magnitudes of the vectors 3 i -2 j , 2 i +2 j + k , 4 i - j + k , 2 i +2 j +3 k . Which one of the following is correct?

NDA 2011 Solution View → 81

What is the projection of the vector i -2 j + k on the vector 4 i -4 j +7 k ?

NDA 2011 Solution View → 82

Which one of the following is the unit vector perpendicular to the vector 4 i +2 j and -3 i +2 j ?

NDA 2011 Solution View → 83

What is the sine of angle between the vectors i +2 j +3 k and - i +2 j +3 k ?

NDA 2011 Solution View → 84

The points with position vectors 10 i +3 j , 12 i -5 j , a i +11 j are collinear, if the value of a is

NDA 2011 Solution View → 85

Consider the following statements in respect of the vectors aligned & u ₁=(1,2,3), u ₂=(2,3,1), & u ₃=(1,3,2), and u ₄=(4,6,2) aligned 1. u ₁ is parallel to u ₄ 2. u ₂ is parallel

NDA 2011 Solution View → 86

What is the area of a triangle whose vertices are at (3,-1,2),(1,-1,-3) and (4,-3,1) ?

NDA 2010 Solution View → 87

The vector a =x i +y i +z k , b = k , c are such that they form a right-handed system. What is c equal to?

NDA 2010 Solution View → 88

What is the value of b such that the scalar product of the vector i + j + k with the unit vector parallel to the sum of the vectors 2 i +4 j -5 k and b i +2 j +3 k is unity?

NDA 2010 Solution View → 89

Consider the following statements: I. For any three vectors a , b , c ; a [( b + c ) ( a + b + c )]=0 II. For any three coplanar unit vectors d , e , f ;( d e ) f =1 Which of the s

NDA 2010 Solution View → 90

If the angle between the vector a and b is 3 , what is the angle between -5 a and 6 b ?

NDA 2010 Solution View → 91

What is the geometric interpretation of the identity ( a - b ) ( a + b )=2( a b ) ? I. If the diagonals of a given parallelogram are used as sides of a second parallelogram, then t

NDA 2010 Solution View → 92

Let a and b be two unit vectors and be the angle between them. If ( a + b ) is also the unit vector, then what is the value of ?

NDA 2010 Solution View → 93

What is the vector equally inclined to the vectors i +3 j and 3 i + j ?

NDA 2010 Solution View → 94

aligned & If & a = i - k & b =x i + j +(1-x) k & c =y i +x j +(1+x-y) k aligned then a ( b c ) depends on

NDA 2010 Solution View → 95

Let a, b and c be distinct non-negative numbers. If the vectors a i +a j +c k , i + k , c i +c j +b k lie on a plane, then which one of the following is correct?

NDA 2010 Solution View → 96

A vector b is collinear with the vector a =(2,1,-1) and satisfies the condition a b =3 . What is b equal to?

NDA 2010 Solution View → 97

If the position vector of a point P with respect to origin O is i +3 j -2 k and that of a point Q is 3 i + j -2 k , then what is the position vector of the bisector of the angle P

NDA 2010 Solution View → 98

What is the value of for which the vectors i - j + k , 2 i + j - k , i - j + k , are coplanar?

NDA 2010 Solution View → 99

P Q R S is a parallelogram, where P Q =3 i +2 j -m k , P S = i +3 j + k and the area of the parallelogram is 90 . What is the value of m ?

NDA 2010 Solution View → 100

A B C D is a quadrilateral. Forces A B , C B , C D and D A act along its sides. What is their resultant?

NDA 2010 Solution View →
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