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Vector Algebra — JEE Main & Advanced Mathematics PYQs

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

616 questionsMathematicsSolutions on every page
1

Let a , b be two vectors, and let P, Q and R be the points with position vectors a , b and a + b , respectively, with respect to the origin O . If | a + b | = 21 , | a - b | = 3 ,

JEE Advanced 2026 Solution View → 2

Let a = 4 i - j + 3 k , b = 10 i + 2 j - k and a vector c be such that 2( a b ) + 3( b c ) = 0 . If a c = 15 , then c ( i + j -3 k ) is equal to:

JEE Main 2026 Solution View → 3

Let a = 2 i + 3 j + 3 k and b = 6 i + 3 j + 3 k . Then the square of the area of the triangle with adjacent sides determined by the vectors (2 a + 3 b ) and ( a - b ) is :

JEE Main 2026 Solution View → 4

If a = i + j + k , b = j - k and c be three vectors such that a c = b and a c = 3 , then c ( a - 2 b ) is equal to _______.

JEE Main 2026 Solution View → 5

Let O be the origin, OP = a and OQ = b . If R is the point on OP such that OP = 5 OR , and M is the point such that OQ = 5 RM , then PM is equal to :

JEE Main 2026 Solution View → 6

Let a = 7 i + j - k and b = j + 2 k . If r is a vector such that r a + a b = 0 and r a = 0 , then |3 r |^2 is equal to:

JEE Main 2026 Solution View → 7

Let u and v be unit vectors inclined at an acute angle such that | u v |= 3 2 . If A = u + v +( u v ) , then is equal to:

JEE Main 2026 Solution View → 8

Let a_k = ( _k) i + j and b_k = i - ( _k) j , where _k = 2^ k-1 2^n + 1 , for some n N , n > 5 . Then the value of _ k=1 ^ n | a_k |^2 _ k=1 ^ n | b_k |^2 is _____.

JEE Main 2026 Solution View → 9

Let the vectors a = - i + j + 3 k and b = i + 3 j + k . For some , R , let c = a + b . If c (3 i - 6 j + 2 k ) = 10 and c ( i + j + k ) = -2 , then | c |^2 is equal to:

JEE Main 2026 Solution View → 10

Two adjacent sides of a parallelogram PQRS are given by PQ = j + k and PS = i - j . If the side PS is rotated about the point P by an acute angle in the plane of the parallelogram

JEE Main 2026 Solution View → 11

If a and b are two vectors such that | a | = 2 and | b | = 3 , then the maximum value of 3 | (3 a + 2 b ) | + 4 | (3 a - 2 b ) | is :

JEE Main 2026 Solution View → 12

Let P be a point in the plane of the vectors A B =3 i + j - k and A C = i - j +3 k such that P is equidistant from the lines AB and AC. If | AP |= 5 2 , then the area of the triang

JEE Main 2026 Solution View → 13

For three unit vectors a , b , c satisfying | a - b |²+| b - c |²+| c - a |²=9 and |2 a +k b +k c |=3 , the positive value of k is

JEE Main 2026 Solution View → 14

Let P Q R be a triangle such that P Q =-2 i - j +2 k and PR =a i +b j -4 k , a, b Z . Let S be the point on QR, which is equidistant from the lines PQ and PR. If | PR |=9 and PS =

JEE Main 2026 Solution View → 15

Let a =2 i -5 j +5 k and b = i - j +3 k . If c is a vector such that 2( a c )+3( b c )= 0 and ( a - b ) c =-97 , then | c k |² is equal to

JEE Main 2026 Solution View → 16

Let a =2 i - j - k , b = i +3 j - k and c =2 i + j +3 k . Let v be the vector in the plane of the vectors a and b , such that the length of its projection on the vector c is 1 14 .

JEE Main 2026 Solution View → 17

Let a =2 i + j -2 k , b = i + j and c = a b . Let d be a vector such that | d - a |= 11 ,| c d |=3 and the angle between c and d is 4 . Then a d is equal to

JEE Main 2026 Solution View → 18

Let a = i -2 j +3 k , b =2 i + j - k , c = i + j + k and v = a b . If v c =11 and the length of the projection of b on c is p , then 9 p² is equal to

JEE Main 2026 Solution View → 19

Let a , b , c be three vectors such that a b =2( a c ) . If | a |=1,| b |=4,| c |=2 , and the angle between b and c is 60^ , then | a c | is equal to

JEE Main 2026 Solution View → 20

Let a =- i + j +2 k , ~b = i - j -3 k , c = a b and d = c a . Then ( a - b ) d is equal to :

JEE Main 2026 Solution View → 21

Let a =2 i - j + k and b = j +2 k , Z be two vectors. Let c = a b and d be a vector of magnitude 2 in y z -plane. If | c |= 53 , then the maximum possible value of ( c d )² is equa

JEE Main 2026 Solution View → 22

Let a vector a = 2 i - j + k , >0 , make an obtuse angle with the vector b =- ² i +4 2 j +4 2 k and an angle , 6 < < 2 , with the positive z -axis. If the set of all possible value

JEE Main 2026 Solution View → 23

Let AB =2 i +4 j -5 k and AD = i +2 j + k , R . Let the projection of the vector v = i + j + k on the diagonal AC of the parallelogram ABCD be of length one unit. If , , where > ,

JEE Main 2026 Solution View → 24

For a triangle ABC, let p = BC , q = CA and r = BA . If | p |=2 3 ,| q |=2 and = 1 3 , where is the angle between p and q , then | p ( q -3 r )|²+3| r |² is equal to :

JEE Main 2026 Solution View → 25

Let a =- i +2 j +2 k , ~b =8 i +7 j -3 k and c be vector such that a c = b . If c ( i + j + k )=4 , then | a + c |² is equal to :

JEE Main 2026 Solution View → 26

For any two points M and N in the X Y -plane, let M N denote the vector from M to N , and 0 denote the zero vector. Let P, Q and R be three distinct points in the X Y -plane. Let S

JEE Advanced 2025 Solution View → 27

Let w = i + j -2 k , and u and v be two vectors such that u v = w and v w = u . Let , , and t be real numbers such that u = i + j + k ,-t + + =0, -t + =0 , and + -t =0 . Match each

JEE Advanced 2025 Solution View → 28

Consider the vectors x = i +2 j +3 k , y =2 i +3 j + k , and z =3 i + j +2 k For two distinct positive real numbers and , define X = x + y - z , Y = y + z - x , and Z = z + x - y I

JEE Advanced 2025 Solution View → 29

Let a = i +2 j + k and b =2 i + j - k . Let c be a unit vector in the plane of the vectors a and b and be perpendicular to a . Then such a vector c is :

JEE Main 2025 Solution View → 30

Let a and b be the vectors of the same magnitude such that | a + b |+| a - b | | a + b |-| a - b | = 2 +1 . Then | a + b |^2 | a |^2 is :

JEE Main 2025 Solution View → 31

Let the angle , 0 2 between two unit vectors a and b be ⁻¹ ( 65 9 ) . If the vector c =3 a +6 ~b +9( a b ), then the value of 9( c a )-3( c b ) is

JEE Main 2025 Solution View → 32

Let the three sides of a triangle A B C be given by the vectors 2 i - j + k , i -3 j -5 k and 3 i -4 j -4 k . Let G be the centroid of the triangle A B C . Then 6 (| AG |^2+| BG |^

JEE Main 2025 Solution View → 33

Consider two vectors u =3 i - j and v =2 i + j - k , 0 . The angle between them is given by ⁻¹ ( 5 2 7 ) . Let v = v ₁+ v ₂ , where v ₁ is parallel to u and v ₂ is perpendicular to

JEE Main 2025 Solution View → 34

Let a = i +2 j + k , b =3 i -3 j +3 k , c =2 i - j +2 k and d be a vector such that b d = c d and a d =4 . Then |( a d )|^2 is equal to ______ .

JEE Main 2025 Solution View → 35

Let a = i + j + k , b =3 i +2 j - k , c = j + k and d be a unit vector such that a d = b d and c d =1 , If c is perpendicular to a , then |3 d + c |^2 is equal to _______ .

JEE Main 2025 Solution View → 36

Let a =2 i -3 j +k, b =3 i +2 j +5 k and a vector c be such that ( a - c ) b =-18 i -3 j +12 k and a c =3 . If b c = d , then | a d | is equal to :

JEE Main 2025 Solution View → 37

Let A B C D be a tetrahedron such that the edges AB , AC and AD are mutually perpendicular. Let the areas of the triangles ABC , ACD and ADB be 5,6 and 7 square units respectively.

JEE Main 2025 Solution View → 38

If a is nonzero vector such that its projections on the vectors 2 i - j +2 k , i +2 j -2 k and k are equal, then a unit vector along a is:

JEE Main 2025 Solution View → 39

Let a be a unit vector perpendicular to the vectors b = i -2 j +3 k and c =2 i +3 j - k , and makes an angle of ⁻¹ (- 1 3 ) with the vector i + j + k . If a makes an angle of 3 wit

JEE Main 2025 Solution View → 40

Let ( a =2 i - j +3 k , ~b =3 i -5 j + k ) and ( c ) be a vector such that ( a c = c b ) and (( a + c ) ( b + c )=168 ). Then the maximum value of (| c |^2 ) is :

JEE Main 2025 Solution View → 41

If the components of a = i + j + k along and perpendicular to b =3 i + j - k respectively, are 16 11 (3 i + j - k ) and 1 11 (-4 i -5 j -17 k ) , then ^2+ ^2+ ^2 is equal to :

JEE Main 2025 Solution View → 42

Let A, B, C be three points in x y -plane, whose position vector are given by 3 i + j , i + 3 j and a i +(1- a ) j respectively with respect to the origin O . If the distance of th

JEE Main 2025 Solution View → 43

Let a = i + j + k , b =2 i +2 j + k and d = a b . If c is a vector such that a c =| c | , | c -2 a |^2=8 and the angle between d and c is 4 , then |10-3 ~b c |+| d c |^2 is equal t

JEE Main 2025 Solution View → 44

Let the position vectors of three vertices of a triangle be 4 p + q -3 r ,-5 p + q +2 r and 2 p - q +2 r . If the position vectors of the orthocenter and the circumcenter of the tr

JEE Main 2025 Solution View → 45

Let a =3 i - j +2 k , ~b = a ( i -2 k ) and c = b k . Then the projection of c -2 j on a is :

JEE Main 2025 Solution View → 46

Let a = i +2 j +3 k , b =3 i + j - k and c be three vectors such that c is coplanar with a and b . If the vector C is perpendicular to b and a c =5 , then | c | is equal to

JEE Main 2025 Solution View → 47

Let the position vectors of the vertices A, B and C of a tetrahedron A B C D be i +2 j + k , i +3 j -2 k and 2 i + j - k respectively. The altitude from the vertex D to the opposit

JEE Main 2025 Solution View → 48

Let a and b be two unit vectors such that the angle between them is 3 . If a +2 b and 3 a - b are perpendicular to each other, then the number of values of in [-1,3] is :

JEE Main 2025 Solution View → 49

Let c be the projection vector of b = i +4 k , 0 , on the vector a = i +2 j +2 k . If | a + c |=7 , then the area of the parallelogram formed by the vectors b and c is ________

JEE Main 2025 Solution View → 50

Let p =2 i + j +3 k and q = i - j + k . If for some real numbers , , and , we have 15 i +10 j +6 k = (2 p + q )+ ( p -2 q )+ ( p q ) then the value of is

JEE Advanced 2024 Solution View → 51

Let O P = -1 i + j + k , O Q = i + -1 j + k and O R = i + j + 1 2 k be three vectors, where , R - 0 and O denotes the origin. If ( O P O Q ) O R =0 and the point ( , , 2) lies on t

JEE Advanced 2024 Solution View → 52

Between the following two statements: Statement I : Let a = i +2 j -3 k and b =2 i + j - k . Then the vector r satisfying a r = a b and a r =0 is of magnitude 10 . Statement II : I

JEE Main 2024 Solution View → 53

Let a =2 i + j + k , b =- i + k , c = j - k , where and are integers and =-6 . Let the values of the ordered pair ( , ) , for which the area of the parallelogram of diagonals a + b

JEE Main 2024 Solution View → 54

Let three vectors a = i +4 j +2 k , b =5 i +3 j +4 k , c =x i +y j +z k form a triangle such that c = a - b and the area of the triangle is 5 6 . If is a positive real number, then

JEE Main 2024 Solution View → 55

Let O A =2 a , O B =6 a +5 b and O C =3 b , where O is the origin. If the area of the parallelogram with adjacent sides OA and OC is 15 sq. units, then the area (in sq. units) of t

JEE Main 2024 Solution View → 56

Let a =4 i - j + k , b =11 i - j + k and c be a vector such that ( a + b ) c = c (-2 a +3 b ) . If (2 a +3 b ) c =1670 , then | c |^2 is equal to :

JEE Main 2024 Solution View → 57

Let a = i +2 j +3 k , b =2 i +3 j -5 k and c =3 i - j + k be three vectors. Let r be anit vector along b + c . If r a =3 , then 3 is equal to:

JEE Main 2024 Solution View → 58

The set of all , for which the vectors a = t i +6 j -3 k and b =t i -2 j -2 t k are inclined at an obtuse angle for all t R , is

JEE Main 2024 Solution View → 59

Let a =9 i -13 j +25 k , b =3 i +7 j -13 k and c =17 i -2 j + k be three given vectors. If r is a vector such that r a =( b + c ) a and r ( b - c )=0 , then |593 r +67 a |^2 (593)^

JEE Main 2024 Solution View → 60

Let a =2 i + j - k , b =(( a ( i + j )) i ) i . Then the square of the projection of a on b is :

JEE Main 2024 Solution View → 61

Let a =6 i + j - k and b = i + j . If c is a is vector such that | c | 6, a c =6| c |,| c - a |=2 2 and the angle between a b and c is 60^ , then |( a b ) c | is equal to:

JEE Main 2024 Solution View → 62

Let a =2 i -3 j +4 k , b =3 i +4 j -5 k and a vector c be such that a ( b + c )+ b c = i +8 j +13 k . If a c =13 , then (24- b c ) is equal to_______

JEE Main 2024 Solution View → 63

Consider three vectors a , b , c . Let | a |=2,| b |=3 and a = b c . If [0, 3 ] is the angle between the vectors b and c , then the minimum value of 27| c - a |^2 is equal to:

JEE Main 2024 Solution View → 64

Let a =2 i +5 j - k , b =2 i -2 j +2 k and c be three vectors such that ( c + i ) ( a + b + i )= a ( c + i ) . If a c =-29 , then c (-2 i + j + k ) is equal to:

JEE Main 2024 Solution View → 65

Let a = i -3 j +7 k , b =2 i - j + k and c be a vector such that ( a +2 b ) c =3( c a ) . If a c =130 , then b c is equal to _______

JEE Main 2024 Solution View → 66

Let a = i + j + k , b =2 i +4 j -5 k and c =x i +2 j +3 k , x R . If d is the unit vector in the direction of b + c such that a d =1 , then ( a b ) c is equal to

JEE Main 2024 Solution View → 67

For >0 , let be the angle between the vectors a = i + j -3 k and b =3 i - j +2 k . If the vectors a + b and a - b are mutually perpendicular, then the value of (14 cos )^2 is equal

JEE Main 2024 Solution View → 68

Let a unit vector which makes an angle of 60^ with 2 i +2 j - k and angle 45^ with i - k be C . Then C + (- 1 2 i + 1 3 2 j - 2 3 k ) is :

JEE Main 2024 Solution View → 69

Let ABC be a triangle of area 15 2 and the vectors AB = i +2 j -7 k , BC = a i + b j + ck and AC =6 i + d j -2 k , ~d >0 . Then the square of the length of the largest side of the

JEE Main 2024 Solution View → 70

Consider a Δ A B C where A 1 , 3 , 2 , B − 2 , 8 , 0 and C 3 , 6 , 7 . If the angle bisector of ∠ B A C meets the line B C at D , then the length of the projection of the vector A

JEE Main 2024 Solution View → 71

Let a → = i ^ + j ^ + k ^ , b → = − i ^ − 8 j ^ + 2 k ^ and c → = 4 i ^ + c 2 j ^ + c 3 k ^ be three vectors such that b → × a → = c → × a → . If the angle between the vector c → a

JEE Main 2024 Solution View → 72

Let a → = − 5 i ^ + j ^ − 3 k ^ , b → = i ^ + 2 j ^ − 4 k ^ and c → = a → × b → × i ^ × i ^ × i ^ . Then c → ⋅ − i ^ + j ^ + k ^ is equal to

JEE Main 2024 Solution View → 73

Let a → = 3 i ^ + 2 j ^ + k ^ , b → = 2 i ^ − j ^ + 3 k ^ and c → be a vector such that a → + b → × c → = 2 a → × b → + 24 j ^ − 6 k ^ and a → − b → + i ^ . c → = − 3 . Then c → 2

JEE Main 2024 Solution View → 74

Let a → = 3 i ^ + j ^ − 2 k ^ , b → = 4 i ^ + j ^ + 7 k ^ and c → = i ^ − 3 j ^ + 4 k ^ be three vectors. If a vectors p → satisfies p → × b → = c → × b → and p → ⋅ a → = 0 , then

JEE Main 2024 Solution View → 75

Let a → and b → be two vectors such that a → = 1 , b → = 4 and a → ⋅ b → = 2 . If c → = 2 a → × b → − 3 b → and the angle between b → and c → is α , then 192 sin 2 α is equal to __

JEE Main 2024 Solution View → 76

Let a → = i ^ + α j ^ + β k ^ , α , β ∈ R . Let a vector b → be such that the angle between a → and b → is π 4 and b → 2 = 6 , If a → · b → = 3 2 , then the value of α 2 + β 2 | a

JEE Main 2024 Solution View → 77

Let a → and b → be two vectors such that | b → | = 1 and | b → × a → | = 2 Then | ( b → × a → ) - b → | 2 is equal to

JEE Main 2024 Solution View → 78

Let A ( 2 , 3 , 5 ) and C ( - 3 , 4 , - 2 ) be opposite vertices of a parallelogram A B C D if the diagonal B D → = i ^ + 2 j ^ + 3 k ^ then the area of the parallelogram is equal

JEE Main 2024 Solution View → 79

Let a → = a i i ^ + a 2 j ^ + a 3 k ^ and b → = b 1 i ^ + b 2 j ^ + b 3 k ^ be two vectors such that | a → | = 1 ; a → · b → = 2 and | b → | = 4 . If c → = 2 ( a → × b → ) - 3 b →

JEE Main 2024 Solution View → 80

Let OA → = a → , OB → = 12 a → + 4 b → and OC → = b → , where O is the origin. If S is the parallelogram with adjacent sides OA and OC , then area of the quadrilateral OABC area of

JEE Main 2024 Solution View → 81

Let a unit vector u ^ = x i ^ + y j ^ + z k ^ make angles π 2 , π 3 and 2 π 3 with the vectors 1 2 i ^ + 1 2 k ^ , 1 2 j ^ + 1 2 k ^ and 1 2 i ^ + 1 2 j ^ respectively. If v → = 1

JEE Main 2024 Solution View → 82

Let a → , b → and c → be three non-zero vectors such that b → and c → are non-collinear if a → + 5 b → is collinear with c → , b → + 6 c → is collinear with a → and a → + α b → + β

JEE Main 2024 Solution View → 83

If a → = i ^ + 2 j ^ + k ^ , b → = 3 i ^ - j ^ + k ^ and c → be the vector such that a → × c → = b → and a → · c → = 3 , then a → · ( c → × b → - b → - c → ) is equal to

JEE Main 2024 Solution View → 84

The least positive integral value of α , for which the angle between the vectors α i ^ - 2 j ^ + 2 k ^ and α i ^ + 2 α j ^ - 2 k ^ is acute, is _____.

JEE Main 2024 Solution View → 85

Let the position vectors of points P , Q , R and S be a &#8594; = i ^ + 2 j ^ - 5 k ^ , b &#8594; = 3 i ^ + 6 j ^ + 3 k ^ , c &#8594; = 17 5 i ^ + 16 5 j ^ + 7 k ^ and d &#8594; =

JEE Advanced 2023 Solution View → 86

Let P be the plane 3 x + 2 y + 3 z = 16 and let S = &#945; i ^ + &#946; j ^ + &#947; k ^ : &#945; 2 + &#946; 2 + &#947; 2 = 1 and the distance of &#945; , &#946; , &#947; from the

JEE Advanced 2023 Solution View → 87

Let S be the set of all &#955; , &#956; for which the vectors &#955; i ^ - j ^ + k ^ , j ^ + 2 j ^ + &#956; k ^ and 3 i ^ - 4 j ^ + 5 k ^ , where &#955; - &#956; = 5 , are coplanar

JEE Main 2023 Solution View → 88

Let A B C D be a quadrilateral. If E and F are the mid points of the diagonals A C and B D respectively and A B &#8594; - B C &#8594; + A D &#8594; - D C &#8594; = k F E &#8594; ,

JEE Main 2023 Solution View → 89

Let | a &#8594; | = 2 , | b &#8594; | = 3 and the angle between the vectors a &#8594; and b &#8594; be &#960; 4 . Then | ( a &#8594; + 2 b &#8594; ) &#215; ( 2 a &#8594; - 3 b &#85

JEE Main 2023 Solution View → 90

Let for a triangle A B C A B &#8594; = - 2 i ^ + j ^ + 3 k ^ C B &#8594; = &#945; i ^ + &#946; j ^ + &#947; k ^ C A &#8594; = 4 i ^ + 3 j ^ + &#948; k ^ If &#948; &#62; 0 and the a

JEE Main 2023 Solution View → 91

Let a &#8594; = i ^ + 4 j ^ + 2 k ^ , b &#8594; = 3 i ^ - 2 j ^ + 7 k ^ and c &#8594; = 2 i ^ - j ^ + 4 k ^ . If a vector d &#8594; satisfies d &#8594; &#215; b &#8594; = c &#8594;

JEE Main 2023 Solution View → 92

Let a &#8594; = 3 i ^ + j ^ - k ^ and c &#8594; = 2 i ^ - 3 j ^ + 3 k ^ . If b &#8594; is a vector such that a &#8594; = b &#8594; &#215; c &#8594; and | b &#8594; | 2 = 50 , then

JEE Main 2023 Solution View → 93

Let a , b , c be three distinct real numbers, none equal to one. If the vectors a i ^ + j ^ + k ^ , i ^ + b j ^ + k ^ and i ^ + j ^ + c k ^ are coplanar, then 1 1 - a + 1 1 - b + 1

JEE Main 2023 Solution View → 94

Let &#955; &#8712; &#8484; , a &#8594; = &#955; i ^ + j ^ - k ^ and b &#8594; = 3 i ^ - j ^ + 2 k ^ . Let c &#8594; be a vector such that a &#8594; + b &#8594; + c &#8594; &#215; c

JEE Main 2023 Solution View → 95

Let the plane x + 3 y - 2 z + 6 = 0 meet the co-ordinate axes at the points A , B , C . If the orthocenter of the triangle A B C is &#945; , &#946; , 6 7 , then 98 &#945; + &#946;

JEE Main 2023 Solution View → 96

If four distinct points with position vectors a &#8594; , b &#8594; , c &#8594; and d &#8594; are coplanar, then [ a &#8594; b &#8594; c &#8594; ] is equal to

JEE Main 2023 Solution View → 97

Let a &#8594; = i ^ + 2 j ^ + 3 k ^ and b &#8594; = i ^ + j ^ - k ^ . If c &#8594; is a vector such that a &#8594; &#183; c &#8594; = 11 , b &#8594; &#183; a &#8594; &#215; c &#859

JEE Main 2023 Solution View → 98

For any vector a &#8594; = a 1 i ^ + a 2 j ^ + a 3 k ^ , with 10 a i &#60; 1 , i = 1 , 2 , 3 , consider the following statements: A : max a 1 , a 2 , a 3 &#8804; a &#8594; B : | a

JEE Main 2023 Solution View → 99

Let a &#8594; be a non-zero vector parallel to the line of intersection of the two planes described by i ^ + j ^ , i ^ + k ^ and i ^ - j ^ , j ^ - k ^ . If &#952; is the angle betw

JEE Main 2023 Solution View → 100

Let a &#8594; = 2 i ^ + 7 j ^ - k ^ , b ^ = 3 i ^ + 5 k ^ and c &#8594; = i ^ - j ^ + 2 k ^ Let d &#8594; be a vector which is perpendicular to both a &#8594; and b &#8594; , and c

JEE Main 2023 Solution View → 101

If the points P and Q are respectively the circumcenter and the orthocentre of a &#8710; A B C , then P A &#8594; + P B &#8594; + P C &#8594; is equal to _______

JEE Main 2023 Solution View → 102

An arc P Q of a circle subtends a right angle at its centre O . The mid point of the arc P Q is R . If O P &#8594; = u &#8594; , O R &#8594; = v &#8594; and O Q &#8594; = &#945; u

JEE Main 2023 Solution View → 103

Let O be the origin and the position vector of the point P be - i ^ - 2 j ^ + 3 k . If the position vectors of the points A , B and C are - 2 i ^ + j ^ - 3 k , 2 i ^ + 4 j ^ - 2 k

JEE Main 2023 Solution View → 104

Let the vectors u 1 &#8594; = i ^ + j ^ + a k ^ , u 2 &#8594; = i ^ + b j ^ + k ^ , and u 3 &#8594; = c i ^ + j ^ + k ^ be coplanar. If the vectors v 1 &#8594; = a + b i ^ + c j ^

JEE Main 2023 Solution View → 105

The area of the quadrilateral A B C D with vertices A ( 2 , 1 , 1 ) , B ( 1 , 2 , 5 ) , C ( - 2 , - 3 , 5 ) and D ( 1 , - 6 , - 7 ) is equal to

JEE Main 2023 Solution View → 106

If the points with position vectors &#945; i ^ + 10 j ^ + 13 k ^ , 6 i ^ + 11 j ^ + 11 k ^ , 9 2 i ^ + &#946; j ^ - 8 k ^ are collinear, then 19 &#945; - 6 &#946; 2 is equal to

JEE Main 2023 Solution View → 107

Let a &#8594; = 6 i ^ + 9 j ^ + 12 k ^ , b &#8594; = &#945; i ^ + 11 j ^ - 2 k ^ and c &#8594; be vectors such that a &#8594; &#215; c &#8594; = a &#8594; &#215; b &#8594; If a &#8

JEE Main 2023 Solution View → 108

Let the vectors a &#8594; , b &#8594; , c &#8594; represent three coterminous edges of a parallelopiped of volume V . Then the volume of the parallelopiped, whose coterminous edges

JEE Main 2023 Solution View → 109

The sum of all values of &#945; , for which the points whose position vectors are i ^ - 2 j ^ + 3 k ^ , 2 i ^ - 3 j ^ + 4 k ^ , &#945; + 1 i ^ + 2 k ^ and 9 i ^ + &#945; - 8 j ^ +

JEE Main 2023 Solution View → 110

Let the position vectors of the points A , B , C and D be 5 i ^ + 5 j ^ + 2 &#955; k ^ , i ^ + 2 j ^ + 3 k ^ , - 2 i ^ + &#955; j ^ + 4 k ^ and - i ^ + 5 j ^ + 6 k ^ . Let the set

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Let a &#8594; = 2 i ^ + 3 j ^ + 4 k ^ , b &#8594; = i ^ - 2 j ^ - 2 k ^ and c &#8594; = - i ^ + 4 j ^ + 3 k ^ . If d &#8594; is a vector perpendicular to both b &#8594; and c &#859

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Let a &#8594; = 5 i ^ - j ^ - 3 k ^ and b &#8594; = i ^ + 3 j ^ + 5 k ^ be two vectors. Then which one of the following statements is TRUE?

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Let a &#8594; = 2 i ^ - 7 j ^ + 5 k ^ , b &#8594; = i ^ + k ^ and c &#8594; = i ^ + 2 j ^ - 3 k ^ be three given vectors. If r &#8594; is a vector such that r &#8594; &#215; a &#85

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Let v &#8594; = &#945; i ^ + 2 j ^ - 3 k ^ , w &#8594; = 2 &#945; i ^ + j ^ - k ^ , and u &#8594; be a vector such that u &#8594; = &#945; &#62; 0 . If the minimum value of the sca

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A 2 , 6 , 2 , B - 4 , 0 , &#955; , C 2 , 3 , - 1 and D 4 , 5 , 0 , &#955; &#8804; 5 are the vertices of a quadrilateral A B C D . If its area is 18 square units, then 5 - 6 &#955;

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Let a &#8594; = i ^ + 2 j ^ + 3 k ^ , b &#8594; = i ^ - j ^ + 2 k ^ and c &#8594; = 5 i ^ - 3 j ^ + 3 k ^ , be there(three) vector. If r &#8594; is a vector such that, r &#8594; &#

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Let a &#8594; , b &#8594; , c &#8594; be three vectors such that a &#8594; = 31 , 4 b &#8594; = c &#8594; = 2 and 2 a &#8594; &#215; b &#8594; = 3 c &#8594; &#215; a &#8594; . If t

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Let a &#8594; = 2 i ^ + j ^ + k ^ , and b &#8594; and c &#8594; be two nonzero vectors such that a &#8594; + b &#8594; + c &#8594; = a &#8594; + b &#8594; - c &#8594; and b &#8594;

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Let a &#8594; and b &#8594; be two vector such that a &#8594; = 14 , b &#8594; = 6 and a &#8594; &#215; b &#8594; = 48 . Then a &#8594; &#183; b &#8594; 2 is equal to _____ .

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Let &#955; &#8712; &#8477; , a &#8594; = &#955; i ^ + 2 j ^ - 3 k ^ , b &#8594; = i ^ - &#955; j ^ + 2 k ^ , If ( ( a &#8594; + b &#8594; ) &#215; ( a &#8594; &#215; b &#8594; ) )

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