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NDA Vector Algebra PYQs with Solutions

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

100 questionsMathematics2010–2026Answers + solutions

About Vector Algebra PYQs

Vector Algebra has 100 previous year questions in the Quantrex Academy NDA Mathematics bank. The 100 questions in this list come from NDA (100), spanning 2010 to 2026. Each question opens on its own page with the options, the correct answer and the worked solution where available.

NDA Vector Algebra PYQs with Solutions

About Vector Algebra PYQs

Questions by year (this list)

Vector Algebra questions with solutions

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

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

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

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

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

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

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 ?

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

A force F = 2 i - λ 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 + 2λ k , then what is the value of λ ?

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 ?

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

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?

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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 ?

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?

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

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

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

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 ?

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

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

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

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

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

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 ?

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

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

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

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

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 ?

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?

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

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

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

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

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

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

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

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

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 ?

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

The vectors a ×( b × a ) is coplanar with :

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

The vector 2 j - k lies:

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

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

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

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?

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

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) ?

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 ?

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

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?

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?

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

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) ?

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 ?

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?

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?

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