Cross Product: JEE Questions & Solutions

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

Let c and d be vectors such that |c+d| = √29 and c × (2i+3j+4k) = (2i+3j+4k) × d. If λ₁, λ₂ (λ₁ > λ₂) are the possible values of (c+d) ·…

For a triangle ABC, let p = BC, q = CA and r = BA. If |p| = 2√3, |q| = 2 and cos θ = 1/√3, where θ is the angle between p and q, then |p ×…

Let a = 2i - j + k and b = λ j + 2k, where λ ∈ ℤ, be two vectors. Let c = a × b and d be a vector of magnitude 2 in the yz-plane. If |c| =…

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

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|…

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

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…

Let a=2i-5j+5k and b=i-j+3k. 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

Let PQR be a triangle such that PQ=-2 i- j+2 k, 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…

Let c be the projection vector of b = λ i + 4 k, λ > 0, on the vector a = i + 2 j + 2 k. If |a + c| = 7, then the area of the parallelogram…

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

Let a = i + j + k, b = 2i + 2j + k and d = a × b. If c is a vector such that a · c = |c|, |c - 2a|² = 8 and the angle between d and c is…

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

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

Let ABCD 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…

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

Let a = i + j + k, b = 3i + 2j - k, c = λ j + μ k and d be a unit vector such that a × d = b × d and c · d = 1. If c is perpendicular to a,…

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

Let A be the point of intersection of the lines L₁: (x - 7)/1 = (y - 5)/0 = (z - 3)/(-1) and L₂: (x - 1)/3 = (y + 3)/4 = (z + 7)/5 Let B…

Let the area of the triangle formed by the lines x + 2 = y - 1 = z, (x - 3)/5 = y/(-1) = (z - 1)/1 and x/(-3) = (y - 3)/3 = (z - 2)/1 be A.…

Let a = i + 2j + k and b = 3(i - j + k). Let c be the vector such that a × c = b and a · c = 3. Then a · ((c × b) - b - c) is equal to:

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

Let A(2,3,5) and C(-3,4,-2) be opposite vertices of a parallelogram ABCD. If the diagonal BD = i + 2j + 3k, then the area of the…

Let a = i + α j + β k, where α, β ∈ ℝ. Let b be a vector such that the angle between a and b is π/4 and |b| = 6. If |a × b| = 3√2, then the…

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

Let a = 3i + j - 2k, b = 4i + j + 7k and c = i - 3j + 4k be three vectors. If a vector p satisfies p × b = c × b and p · a = 0, then p · (i…

Let a and b be two vectors such that |a| = 1, |b| = 4 and a · b = 2. If c = (2a × b) - 3b and the angle between b and c is α, then 192…

The area of a quadrilateral ABCD with vertices A(3, 1, -1), B(5/3, 7/3, 1/3), C(2, 2, 1), D(10/3, 2/3, -1/3) is:

Let a = 2i + α j + k, b = -i + k, c = β j - k, where α and β are integers and αβ = -6. Let the values of the ordered pair (α, β) for which…

Between the following two statements: Statement-I: Let a = i + 2j - 3k and b = 2i + j - k. Then the vector r satisfying a × r = a × b and a…

Let a = i + 2j + λ k, b = 3i - 5j - λ k, a · c = 7, 2b · c + 43 = 0, a × c = b × c. Then |a · b| is equal to:

Let a, b, c be three non-zero vectors such that b · c = 0 and a × (b × c) = (b - c)/2. If d is a vector such that b · d = a · b, then (a ×…

If the four points, whose position vectors are 3i - 4j + 2k, i + 2j - k, -2i - j + 3k, and 5i - 2αj + 4k are coplanar, then α is equal to:

Let a = -i - j + k, a · b = 1 and a × b = i - j. Then a - 6b is equal to:

Let the coordinates of one vertex of triangle ABC be A(0, 2, α) and the other two vertices lie on the line (x + α)/5 = (y - 1)/2 = (z +…

Let a, b, and c be three non-zero, non-coplanar vectors. Let the position vectors of four points A, B, C, and D be a - b + c, λ a - 3b +…

If a = i + 2k, b = i + j + k, c = 7i - 3j + 4k, and r × b + b × c = 0 and r · a = 0 then r · c is equal to:

Let a = 4i + 3j, b = 3i - 4j + 5k, and c be a vector such that (a × b) · c + 25 = 0, c · (i + j + k) = 4, and the projection of c on a is…

If a, b, c are three non-zero vectors and n is a unit vector perpendicular to c such that a = α b - n, (α ≠ 0) and b · c = 12, then | c ×…

Let λ ∈ ℝ, a = λ i + 2j - 3k, b = i - λ j + 2k. If ( (a + b) × (a × b)) × (a - b) = 8i - 40j - 24k, then |λ (a + b) × (a - b)|² is equal…

Let a and b be two vectors such that |a| = √14, |b| = √6, |a × b| = √48. Then (a · b)² is equal to:

Let a = i + 2j + 3k, b = i - j + 2k, c = 5i - 3j + 3k be three vectors. If r is a vector such that r × b = c × b and r · a = 0, then 25|r|²…

Let a, b, c be three vectors such that |a| = √31, 4|b| = |c| = 2, 2(a × b) = 3(c × a). If the angle between b and c is 2π/3, then ( (a ×…

Let v = α i + 2 j - 3 k, w = 2α i + j - k, and u be a vector such that |u| = α > 0. If the minimum value of the scalar triple product [u v…

Let a = 2i - 7j + 5k, b = i + k, and c = i + 2j - 3k be three given vectors. If r is a vector such that r × a = c × a and r · b = 0, then…

Let the position vectors of the points A, B, C and D be 5i + 5j + 2λ k, i + 2j + 3k, -2i + λj + 4k and -i + 5j + 6k. Let the set S = {λ ∈…

Let a = 2i + 3j + 4k, b = i - 2j - 2k and c = -i + 4j + 3k. If d is a vector perpendicular to both b and c, and a · d = 18, then |a × d|²…

The sum of all values of α, for which the points whose position vectors are i - 2j + 3k, 2i - 3j + 4k, (α + 1)i + 2k and 9i + (α - 8)j + 6k…

Let the vectors a, b, c represent three coterminous edges of a parallelepiped of volume V. Then the volume of the parallelepiped, whose…

If the points with position vectors α i + 10j + 13k, 6i + 11j + 11k, and 9/2i + βj - 8k are collinear, then (19α - 6β)² is equal to:

Let a = 6i + 9j + 12k, b = αi + 11j - 2k, and c be vectors such that a × c = a × b. If a · c = -12 and c · (i - 2j + k) = 5, then c · (i +…

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

Let a = 2i + 7j - k, b = 3i + 5k, c = i - j + 2k. Let d be a vector which is perpendicular to both a and b, and c · d = 12. The value of (…

Let a be a non-zero vector parallel to the line of intersection of the two planes described by i + j, i + k and i - j, i - k. If θ is the…

If four distinct points with position vectors a, b, c, and d are coplanar, then [abc] is equal to:

Let a = i + 2j + 3k and b = i + j - k. If c is a vector such that a · c = 11, b · (a × c) = 27 and b · c = -√3|b|, then |a × c|² is equal…

Let a, b, c be three distinct real numbers, none equal to one. If the vectors a i + j + k, i + bj + k, i + j + c k are coplanar, then 1/(1…

Let λ ∈ ℤ, a = λ i + j - k and b = 3i - j + 2k. Let c be a vector such that (a + b + c) × c = 0, a · c = -17 and b · c = -20. Then | c × (λ…

Let a = i + 4j + 2k, b = 3i - 2j + 7k, c = 2i - j + 4k. If a vector d satisfies d × b = c × b and d · a = 24, then |d|² is equal to:

Let for a triangle ABC, AB = -2i + j + 3k, CB = αi + βj + γk and CA = 4i + 3j + δk. If δ > 0 and the area of the triangle ABC is 5√6, then…

Let |a| = 2, |b| = 3 and the angle between the vectors a and b be π/4. Then |(a + 2b) × (2a - 3b)|² is equal to:

Let S be the set of all (λ,μ) for which the vectors λ i-j+k, i+2j+μk, and 3i-4j+5k where λ-μ=5, are coplanar. Then Σ_((λ,μ)∈ S) 80(λ²+μ²)…