Physics · Motion In A Plane · NEET
It is a VECTOR. The result A x B has both magnitude (A B sin theta) and a direction (perpendicular to both A and B). This is the key difference from the dot product A . B, which gives just a number. Remember: cross product -> vector, dot product -> scalar (number).
The cross product measures how much the two vectors are perpendicular to each other. When the vectors are at 90 degrees, sin(90) = 1 gives the maximum value. When they are parallel (theta = 0), sin(0) = 0, so the cross product is zero. This is opposite to the dot product, which uses cos and is maximum when vectors point the same way.
Point the fingers of your right hand along the first vector A, then curl them toward the second vector B (through the smaller angle). Your thumb now points in the direction of A x B. In physics this direction is always perpendicular to the flat sheet containing A and B - either out of the page or into the page.
The cross product is anti-commutative: A x B = -(B x A). They have the same magnitude but opposite directions. If A x B points out of the page, then B x A points into the page. This is because reversing the order flips the direction your thumb points in the right-hand rule. (Note: the dot product IS commutative, A . B = B . A.)
It is zero. If A and B point in the same direction (theta = 0) or exactly opposite (theta = 180), then sin(theta) = 0, so A x B = 0 (the null vector). A quick test for NEET: if the cross product of two vectors is zero, the vectors are parallel or anti-parallel. Also, A x A = 0 always.
Use the determinant. For A = Ax i + Ay j + Az k and B = Bx i + By j + Bz k: A x B = (Ay Bz - Az By) i - (Ax Bz - Az Bx) j + (Ax By - Ay Bx) k. Remember the unit-vector cycle: i x j = k, j x k = i, k x i = j (forward = positive). Going backward gives a minus sign: j x i = -k.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
The magnitude |A x B| = A B sin(theta) equals the AREA of the parallelogram formed by A and B as its two sides. Half of it (1/2 |A x B|) is the area of the triangle formed by the two vectors.
Torque (tau = r x F), angular momentum (L = r x p), the magnetic force on a moving charge (F = q v x B), and linear velocity in rotation (v = omega x r). This is why the cross product matters a lot in NEET mechanics and magnetism.
No. In general A x (B x C) is not equal to (A x B) x C, so the cross product is neither commutative nor associative. It is, however, distributive over addition: A x (B + C) = A x B + A x C.
For the standard unit vectors: i x j = k, j x k = i, k x i = j (all positive, following the forward cycle). Reversing the order gives a negative sign, e.g. j x i = -k. Also i x i = j x j = k x k = 0.
Dot product A . B = A B cos(theta) gives a scalar (used for work W = F . d). Cross product A x B = A B sin(theta) n gives a vector (used for torque and magnetic force). Dot is maximum for parallel vectors; cross is maximum for perpendicular vectors.
