Cross Product: JEE Questions & Solutions

Let a = - i + 2 j + 2 k, b = 8 i + 7 j - 3 k 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 (2 i+3 j+4 k) = (2 i+3 j+4 k) d. If 1, 2 ( 1 2) are the possible values of ( c+ d) (-7…

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 -…

Let a = 2 i - j + k and b = j + 2 k, where Z, be two vectors. Let c = a b and d be a vector of magnitude 2 in the yz-plane. If | c| = 53,…

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

Let a, b, c be vectors such that a b = 2( a c). | a| = 1, | b| = 4, | c| = 2, angle between b, c is 60^. Find | b - 2 c|.

Let a, b, c be defined. v = a b. v c = 11. Projection of b on c is p. Find 9p^2.

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.…

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

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 = i + 4 k, 0, on the vector a = i + 2 j + 2 k. If | a + c| = 7, then the area of the parallelogram…

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:

Let a = i + j + k, b = 2i + 2j + k and d = a b. If c is a vector such that a c = | c|, c - 2 d = 8 and the angle between d and c is 4, then…

Let a = 2 i - j + 3 k, b = 3 i - 5 j + k, and c be a vector such that a c = a 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 - 2 j + 3 k and c = 2 i + 3 j - k, and makes an angle of ( - 13) 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 = 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 = a, then | a c|…

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…

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

Let A be the point of intersection of the lines L1: x - 71 = y - 50 = z - 3-1 and L2: x - 13 = y + 34 = z + 75 Let B and C be the points on…

Let the area of the triangle formed by the lines x + 2-3 = y - 33 = z - 21, x - 35 = y-1 = z - 11 be A. Then A^2 is equal to:

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

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 + 2 j + 3 k, then the area of the…

Let a = i + j + k, where, R. Let b be a vector such that the angle between a and b is /4 and | b| = 6. If | a b| = 3 2, then the value of (…

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 = 3 i + j - 2 k, b = 4 i + j + 7 k and c = i - 3 j + 4 k 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 = (2 a b) - 3 b and the angle between b and c is, then 192 ^2()…

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

The area of a quadrilateral ABCD with vertices A(3, 1, -1), B ( 53, 73, 13), C(2, 2, 1), D ( 103, 23, - 13) is:

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 r =…

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…

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 r =…

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…

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…

Let a = i + 2 j + k, b = 3 i - 5 j - k, a c = 7, 2 b 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 = b - c2. If d is a vector such that b d = a b, then ( a b) ( c d) is equal…

If P = 3 i + 3 j + 2 k and Q = 4 i + 3 j + 2.5 k, the unit vector in the direction of P Q is 1x ( 3 i + j - 2 3 k). The value of x is:

If the four points, whose position vectors are 3 i - 4 j + 2 k, i + 2 j - k, -2 i - j + 3 k, and 5 i - 2 j + 4 k are coplanar, then is…

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

Let the coordinates of one vertex of triangle ABC be A(0, 2,) and the other two vertices lie on the line x + /5 = (y - 1)/2 = (z + 4)/3.…

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, a - 3 b + 4…

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

Let a = 4 i + 3 j, b = 3 i - 4 j + 5 k, 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 = b - n, ( 0) and b c = 12, then | c ( a b) |…

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

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

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

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

Let a = 2 i - 7 j + 5 k, b = i + k, and c = i + 2 j - 3 k 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 5 i + 5 j + 2 k, i + 2 j + 3 k, -2 i + x j + 4 k and - i + 5 j + 6 k. Let the set S…

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

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

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 i + 10 j + 13 k, 6 i + 11 j + 11 k, and 92 i + j - 8 k are collinear, then (19 - 6)^2 is equal to:

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

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

Let a = 2 i + 7 j - k, b = 3 i + 5 k, c = i - j + 2 k. 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 + k and -i - j - k. If is the angle…

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

Let a = i + 2 j + 3 k 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 to:

Let a, b, c be three distinct real numbers, none equal to one. If the vectors a i + j + k, i + b j + k, a i + j + c k are coplanar, then 11…

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

Let a = i + 4 j + 2 k, b = 3 i - 2 j + 7 k, c = 2 i - j + 4 k. 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 = -2 i + j + 3 k, CB = i + j + k, CA = 4 i + 3 j + 4 k If = 0 and the area of triangle ABC is 5 6, then CB is…

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

Let S be the set of all (,) for which the vectors i- j+ k, i+2 j+ k, and 3 i-4 j+5 k where - =5, are coplanar. Then (,) S 80( ^2+ ^2) is…