2.9

Vector Product (Cross Product)

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The cross or vector product of two vectors is the product of their magnitudes and the sine of the angle. It is directed perpendicular to the vectors' plane.

Geometrically, it is the area of the parallelogram spanned by the vectors.

Using the right-hand rule, if the index finger and the middle fingers of the right-hand point the first and the second vectors, respectively, the thumb finger points to their cross product.

The cross product of the second vector with the first has the opposite direction. Hence, order matters for cross products.

The measure of a door's rotation results from a torque which is the cross product of the radial distance between the hinge and the point of application of force with the applied force.

The door does not rotate if a force acts along or opposite the direction of radial distance.

For the unit vector of an axis, its cross product with itself is zero. But with another, is the third one in cyclic-order.

The components of the cross product of vectors A and B are A_yB_z minus A_zB_y, A_zB_x minus A_xB_z, and A_xB_y minus A_yB_x.

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$$\rightleftharpoonup{xx}$$ $$\longleftharp{xx}$$, $$\longrightharp{xx}$$,

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.

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Vector Multiplication

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