Vector multiplication of two vectors yields a vector product, with the magnitude equal to the product of the individual vectors multiplied by the sine of the angle between both the vectors and the direction perpendicular to both the individual vectors. As there are always two directions perpendicular to a given plane, one on each side, the direction of the vector product is governed by the right-hand thumb rule.
Consider the cross product of two vectors. Imagine rotating the first vector about the perpendicular line until it is aligned with the second, choosing the smaller of the two possible angles between the two vectors. The fingers of the right hand are curled around the perpendicular line so that the fingertips point in the direction of rotation; the thumb then points in the direction of the cross product. Similarly, the direction of the cross product of the second vector with the first is determined by rotating the second vector into the first vector, and it is opposite to the cross product of the first vector with the second. This implies that these two are antiparallel to each other. The cross product of two vectors is anti-commutative, which means that the vector product reverses the sign when the order of multiplication is reversed. The vector product of two parallel or antiparallel vectors is always zero. In particular, the vector product of any vector with itself is zero.
Let us consider a few cases where the vector product is applied. The mechanical advantage that a wrench provides depends on the magnitude of the applied force, on its direction with respect to the wrench handle, and on how far from the nut this force is applied. The physical vector quantity that makes the nut turn is called torque, and it is the vector product of the applied force vector with the position vector. Another example is the case of a particle moving in a magnetic field that experiences a magnetic force. The magnetic field, magnetic force, and velocity are vector quantities. The force vector is proportional to the vector product of the velocity vector with the magnetic field vector.
This text is adapted from Openstax, University Physics Volume 1, Section 2.4: Products of Vectors.
