Vector Product of Two Vectors

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2.5Vector Product of Two Vectors


In these revision notes for Vector Product of Two Vectors, we cover the following key points:

  • The meaning of cross product of two vectors
  • What does the cross product represent geometrically?
  • How to calculate the cross product of two vectors?
  • How to find the direction of cross product vector?
  • How to calculate the cross product in coordinates?
  • Which are some applications of cross product in Physics?

Vector Product of Two Vectors Revision Notes

Symbolically, the cross product of two vectors a and b is denoted through the symbol (×).

Geometrically, it represents a new vector, which is perpendicular to the plane on which the two vectors lie.

Mathematically, the cross product of two vectors represents the magnitude of the surface area enclosed by the two vectors a and b and their parallel extensions (the area of the parallelogram formed by the two vectors a and b).

The cross product of two vectors a and b is

c = a × b

and its magnitude is

|c|=|a×b|

Or

|c| = |a × b| = |a| × |b| × sin θ

where θ is the angle between the vectors a and b.

For the cross product of two vectors, the following rule is true

a × b = -b× a

If the coordinates of the vectors a and b (namely xa, ya, za, xb, yb and zb) are given, we can find the coordinates of the vector c = a × b (i.e. xc, yc and zc) using the following formulae:

xc = ya × zb - yb × za
yc = za × xb - zb × xa
zc = xa × yb - xb × ya

In Physics, there are many applications of vectors cross product. Some of them include:

  • Moment of force M as cross product of Force (F) and linear distance from the turning point (Δx).
  • Magnetic force F of a conductor at rest as a cross product of Magnetic induction (B) and the conductor length L multiplied with the scalar I (current).
  • Magnetic force F of a moving conductor as a cross product of Magnetic induction (B) and Velocity v multiplied with the scalar q (charge).
  • The angle between two forces F1 and F2 can be calculated using the cross product if the magnitudes of the two vectors F1 and F2 and that of F1 × F2 are known.

If we multiply the cross product of two vectors with a scalar, the result is still a vector as the cross product gives a vector and the product of a vector by a scalar gives a vector as well.

In the cross product

c = a× b

we can apply the "right hand rule" to find the direction of the vector product c when the directions of a and b are known. Thus, the index and middle fingers represent the vectors a and b respectively, while the thumb shows the direction of vector product

c = a× b

based on the drill (screwdriver) rule.

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