Vector Algebra

CBSE Class 12 Mathematics · every board question from this chapter, 2022–2026 · 75 of them repeated across years

• The value of p for which vectors i+2 j+3 k and 2 i-p j+ k are perpendicular to each other is2026asked 3×
• The value of m for which the points with position vectors - i- j+2 k, 2 i+m j+5 k and 3 i+11 j+6 k are collinear, is2026asked 3×
• If a =5 and -2 ≤ λ ≤ 1, then the sum of greatest and the smallest value of λ a is2026asked 3×
• Vector of magnitude 3 making equal angles with x and y axes and perpendicular to z axis is2026asked 3×
• For any two vectors a and b, which of the following statements is always true?2026asked 3×
• If a =8, b =3 and a × b =12, then the value of a · b2026asked 3×
• If ( a+ b) ·( a- b)=198 and a =10 b, then:2026asked 3×
• If (3 i-2 j+5 k) ×(4 i+p j+q k)= 0, then the values of p and q are:2026asked 3×
• If position vector p of a point (24, n) is such that p =25, then the value of n is:2026asked 3×
• If vectors a=3 i+2 j+λ k and b=2 i-4 j+5 k, represent the two strips of the Red Cross sign placed outside a doctor's clinic, then…2026asked 3×
• Assertion: The vectors a and (-2 a), where a ≠ 0 are collinear vectors. Reason (R): a ·(-2 a)=0.2026asked 3×
• For two vectors a and b Assertion: a × b ^2+( a · b)^2= a ^2 b ^2 Reason (R): a × b =( a · b) tan θ,(θ ≠ (π)/2)2026asked 3×
• Find the vector of magnitude 14 in the direction of QP, where P and Q are the points (1,3,2) and (-1,0,8) respectively.2026
• Using vectors, find the area of ABC with vertices A(1,2,3), B(2, -1,4) and C(4, 5, -1).2026
• Vectors a=3 i-2 j+2 k and b= i+2 k represent the two adjacent sides of a parallelogram. Find the vectors representing its…2026asked 3×
• Three honey bees were found flying along the vectors a=2 i-3 j+ k, b=4 j-2 k and c=3 i+2 k respectively. Find the value of λ such…2026asked 3×
• If AB= j+ k and AC=3 i- j+4 k represent the two vectors along the sides AB and AC of ABC, prove that the median AD= AB+ AC2,…2026asked 3×
• If the position vectors of three points A, B and C are 3 i+ j, 5 i+6 j-3 k and 4 j respectively, then show that they form an…2026
• If ( 2 i+3 j+ k ) and ( 2 i+ j- k ) represent the sides A B and A C respectively of Δ ABC, find the vector representing the…2026
• If for two unit vectors a and b, a+2 b = 2 a- b, then find the angle between a and b.2026asked 3×
• If in a parallelogram PQRS, PQ=2 i+3 j+ k and QR= i-2 j+ k, then find the unit vectors parallel to the diagonals PR and SQ.2026
• Let two rods placed on the ground be represented by vectors 4 i- j+3 k and -2 i+ j-2 k. Find a vector representing a flag-post of…2026asked 3×
• Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are 55 i-2 j, 5 i+8 j and a…2026asked 3×
• The values of x for which the angle between the vectors a=2 x^2 i+4 x j+ k and b=7 i-2 j+x k is obtuse, is:2025
• If p and q are unit vectors, then which of the following values of p · q is not possible?2025
• The projection vector of vector a on vector b is2025
• If vector a=3 i+2 j- k and vector b= i- j+ k, then which of the following is correct?2025
• The unit vector perpendicular to the vectors i- j and i+ j is2025
• If α= i-4 j+9 k and β=2 i-8 j+λ k are two mutually parallel vectors, then λ is equal to:2025
• If the sides A B and A C of A B C are represented by vectors j+ k and 3 i- j+4 k respectively, then the length of the median…2025asked 3×
• If a+ b+ c= 0, a =√37, b =3 and c =4, then angle between b and c is2025asked 3×
• Let p and q be two unit vectors and α be the angle between them. Then ( p+ q) will be a unit vector for what value of α?2025
• If PQ × PR=4 i+8 j-8 k, then the area ( PQR ) is2025
• If projection of a=α i+ j+4 k on b=2 i+6 j+3 k is 4 units, then α is2025
• A student tries to tie ropes, parallel to each other from one end of the wall to the other. If one rope is along the vector 3…2025asked 3×
• If a+ b = a- b for any two vectors, then vectors a and b are:2025asked 3×
• Let a be a position vector whose tip is the point (2, -3). If AB= a, where coordinates of A are (-4,5), then the coordinates of B…2025asked 3×
• The respective values of a and b, if given ( a- b) ·( a+ b)=512 and a =3 b, are:2025asked 3×
• Assertion: If a × b ^2+ a · b ^2=256 and b =8, then a =2 Reason (R): sin^2 θ+ cos^2 θ=1 and a × b = a b sin θ and a · b= a b cos θ2025asked 3×
• A vector a makes equal angles with all the three axes. If the magnitude of the vector is 5 √3 units, then find a. OR If α and β…2025asked 3×
• The diagonals of a parallelogram are given by a=2 i- j+ k and b= i+3 j- k. Find the area of the parallelogram.2025asked 3×
• If a and b are two non-collinear vectors, then find x, such that α=(x-2) a+ b and β=(3+2 x) a-2 b are collinear.2025
• If a =2, b =3 and a · b=4, then evaluate a+2 b.2025
• Find a vector of magnitude 5 which is perpendicular to both the vectors 3 i-2 j+ k and 4 i+3 j-2 k. OR Let a, b and c be three…2025asked 3×
• If a and b are position vectors of point A and point B respectively, find the position vector of point C on BA produced such that…2025asked 3×
• Let p=2 i-3 j- k, q=-3 i+4 j+ k and r= i+ j+2 k. Express r in the form of r=λ p+μ q and hence find the values of λ and μ.2025
• Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be…2025asked 3×
• If a+ b+ c= 0 such that a =3, b =5, c =7, then find the angle between a and b. OR If a and b are unit vectors inclined with each…2025asked 3×
• The scalar product of the vector a= i- j+2 k with a unit vector along sum of vectors b=2 i-4 j+5 k and c=λ i-2 j-3 k is equal to…2025asked 3×
• Show that the area of a parallelogram whose diagonals are represented by a and b is given by 1/2 a × b. Also find the area of a…2025asked 3×
• Three friends A, B and C move out from the same location O at the same time in three different directions to reach their…2025
• Three friends A, B and C move out from the same location O at the same time in three different directions to reach their…2025asked 2×
• For any two vectors a and b, which of the following statements is always true?2024asked 3×
• Let θ be the angle between two unit vectors a and b such that sin θ=3/5. Then, a · b is equal to:2024asked 3×
• If a=2 i- j+ k and b= i+ j- k, then a and b are:2024asked 2×
• If a and b are two vectors such that a =1, b =2 and a · b=√3, then the angle between 2 a and - b is:2024asked 3×
• The vectors a=2 i- j+ k, b= i-3 j-5 k and c=-3 i+4 j+4 k represents the sides of2024asked 3×
• Let a be any vector such that a =a. The value of a × i ^2+ a × j ^2+ a × k ^2 is:2024asked 3×
• A vector perpendicular to the line r= i+ j- k+λ(3 i- j) is:2024
• If a=2 i- j+ k and b= i+ j- k, then a and b are:2024
• The position vectors of points P and Q are p and q respectively. The point R divides line segment PQ in the ratio 3: 1 and S is…2024asked 3×
• The vector with terminal point A(2,-3,5) and initial point B(3,-4,7) is:2024asked 3×
• If a =2 and -3 ≤ k ≤ 2, then k a ∈:2024asked 3×
• The unit vector perpendicular to both vectors i+ k and i- k is:2024asked 3×
• Assertion: For any non-zero unit vector a, a ·(- a)=(- a) · a=-1. Reason (R): Angle between a and (- a) is (π)/2.2024
• Assertion: Projection of a on b is same as projection of b on a. Reason (R): Angle between a and b is same as angle between b and…2024asked 2×
• Assertion: The vectors represent the sides of a right angled triangle. Reason (R): Three non-zero vectors of which none of two…2024asked 3×
• Assertion: ( b · c) a is a scalar quantity. Reason (R): Dot product of two vectors is a scalar quantity.2024
• Assertion: For two non-zero vectors a and b, a · b= b · a. Reason (R): For two non-zero vectors a and b, a × b= b × a.2024asked 2×
• a, b and c are three mutually perpendicular unit vectors. If θ is the angle between a and ( 2 a+3 b+6 c ), find the value of cos…2024
• If a and b are two non-zero vectors such that ( a+ b) ⊥ a and (2 a+ b) ⊥ b, then prove that b =√2 a.2024asked 3×
• If a, b and c are three unit vectors such that a+ b- c= 0, find the angle between vectors a and c.2024
• Find the position vector of point C which divides the line segment joining points A and B having position vectors i+2 j- k and -…2024asked 3×
• Let a and b be two non-zero vectors. Prove that a × b ≤ a b. State the condition under which equality holds, i.e., a × b = a b.2024
• In the given figure, ABCD is a parallelogram. If AB=2 i-4 j+5 k and D B=3 i-6 j+2 k, then find A D and hence find the area of…2024asked 3×
• If vectors a, b and 2 a+3 b are unit vectors, then find the angle between a and b.2024
• The position vectors of vertices of Δ ABC are A(2 i- j+ k), B( i-3 j-5 k) and C(3 i-4 j-4 k). Find all the angles of Δ ABC.2024asked 2×
• Find a vector of magnitude 4 units perpendicular to each of the vectors 2 i- j+ k and i+ j- k and hence verify your answer.2024
• Given a=2 i- j+ k, b=3 i- k and c=2 i+ j-2 k. Find a vector d which is perpendicular to both a and b and c · d=3.2024
• Find the projection of vector ( b+ c ) on vector a, where a=2 i+2 j+ k, b= i+3 j+ k and c= i+ k.2024
• An instructor at the astronomical centre shows three among the brightest stars in a particular constellation. Assume that the…2024asked 3×
• If the angle between the vectors a and b is (π)/4 and a × b =1, then a · b is equal to2023
• If the angle between a and b is (π)/3 and a × b =3 √3, then the value of a · b is2023
• The position vectors of three consecutive vertices of a parallelogram ABCD are A(4 i+2 j-6 k), B(5 i-3 j+ k) and C(12 i+4 j+5 k).…2023
• If the vector i-b j+ k is equally inclined to the coordinate axes, then the value of b is:2023
• If in ABC, BA=2 a and BC=3 b, then AC is2023
• If a × b =√3 and a · b=-3, then angle between a and b is2023
• If a+ b= i and a=2 i-2 j+2 k, then b equals:2023asked 3×
• Unit vector along PQ, where coordinates of P and Q respectively are (2,1,-1) and (4,4,-7), is2023asked 3×
• The value of p for which the vectors 2 i+p j+ k and -4 i-6 j+26 k are perpendicular to each other, is:2023asked 3×
• The value of ( i × j) · j+( j × i) · k is:2023
• Position vector of the mid-point of line segment AB is 3 i+2 j-3 k. If position vector of the point A is 2 i+3 j-4 k, then…2023
• The sine of the angle between the vectors a=3 i+ j+2 k and b= i+ j+2 k is:2023
• Projection of vector 2 i+3 j on the vector 3 i-2 j is2023
• a and b are two non-zero vectors such that the projection of a on b is 0. The angle between a and b is:2023asked 3×
• In Δ ABC, AB= i+ j+2 k and AC=3 i- j+4 k. If D is mid-point of BC, then vector AD is equal to:2023asked 3×
• A unit vector along the vector 4 i-3 k is:2023asked 3×
• If θ is the angle between two vectors a and b, then a · b ≥ 0 only when:2023asked 2×
• If a · i= a ·( i+ j)= a ·( i+ j+ k)=1, then a is2023
• Two vectors a=a 1 i+a 2 j+a 3 k and b=b 1 i+b 2 j+b 3 k are collinear if2023asked 2×
• The magnitude of the vector 6 i-2 j+3 k is2023asked 2×
• A unit vector a makes equal but acute angles on the co-ordinate axes. The projection of the vector a on the vector b=5 i+7 j- k is2023
• For what value of λ, the projection of vector i+λ j on vector i- j is √2?2023
• If points A, B and C have position vectors 2 i, j and 2 k respectively, then show that ABC is an isosceles triangle.2023
• If r=3 i-2 j+6 k, find the value of ( r × j) ·( r × k)-12.2023asked 3×
• Find all the vectors of magnitude 3 √3 which are collinear to vector i+ j+ k.2023asked 2×
• If a, b, c are three non-zero unequal vectors such that a · b= a · c, then find the angle between a and b- c.2023asked 2×
• Position vectors of the points A, B and C as shown in the figure below are a, b and c respectively. If AC=5/4 AB, express c in…2023asked 3×
• If the projection of the vector i+ j+ k on the vector p i+ j-2 k is 1/3, then find the value(s) of p.2023asked 2×
• If the vectors a and b are such that a =3, b =2/3 and a × b is a unit vector, then find the angle between a and b. OR Find the…2023asked 3×
• For two non-zero vectors a and b, if a- b = a+ b, then find the angle between a and b.2023
• If a=4 i- j+ k and b=2 i-2 j+ k, then find a unit vector along the vector a × b.2023
• Write the projection of the vector ( b+ c) on the vector a, where a= 2 i-2 j+ k, b= i+2 j-2 k and c=2 i- j+4 k.2022asked 3×
• In a parallelogram PQRS, PQ=3 i-2 j+2 k and PS=- i-2 k. Find PR and QS.2022asked 2×
• a and b are two unit vectors such that 2 a+3 b = 3 a-2 b. Find the angle between a and b.2022asked 3×
• If a × b ^2+ a · b ^2=400 and b =5, then find the value of a. OR Find all the possible vectors of magnitude 5 √3 which are…2022asked 3×
• If a= i+ j+ k, a · b=1 and a × b= j- k, then find b2022asked 3×
• If a=2 i+y j+ k and b= i+2 j+3 k are two vectors for which the vector ( a+ b) is perpendicular to the vector ( a- b), then find…2022
• If a, b, c are three vectors such that a · b= a · c and a × b= a × c, a ≠ 0, then show that b= c. OR If a =3, b =5, c =4 and a+…2022
• If a and b are two vectors such that a+ b = b, then prove that ( a+2 b) is perpendicular to a. OR If a and b are unit vectors and…2022asked 2×
• ABCD is a parallelogram such that AC= i+ j and BD=2 i+ j+ k. Find AB and AD. Also, find the area of the parallelogram ABCD.2022
• If a, b, c and d are four non-zero vectors such that a × b= c × d and a × c=4 b × d, then show that ( a-2 d) is parallel to (2 b-…2022asked 3×
• If a=2 i- j+ k, b= i+ j-2 k and c= i+3 j- k and the projection of vector c+λ b on vector a is 2 √6, then find the value of λ.2022
• If a, b and c are mutually perpendicular vectors of equal magnitude, then prove that the vector (2 a+ b+2 c) is equally inclined…2022
• The two adjacent sides of a parallelogram are represented by vectors 2 i-4 j+ 5 k and i-2 j-3 k. Find the unit vector parallel to…2022asked 2×
• Let a= i+ j, b= i- j and c= i+ j+ k. If n is a unit vector such that a. n=0 and b. n=0, then find c. n. OR If a and b are unit…2022
• Using vectors, find the value of 'b' if the points A(-1, -1, 2), B(2, b, 5) and C(3, 11, 6) are collinear. Also, determine the…2022
• If a and b are two vectors of equal magnitude and α is the angle between them, then prove that a+ b a- b = cot ((α)/2).2022
• If a and b are two vectors such that a= i- j+ k and b=2 i- j-3 k, then find the vector c, given that a × c= b and a · c=4.2022

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