Cross Product
Main Concept
The cross product of A⇀ and B⇀ is a vector denoted A⇀ × B⇀. The magnitude of A⇀ × B ⇀ is given by
|A⇀ × B ⇀|= |A⇀ |⋅B⇀⋅sinθ
where θ is the angle between A⇀ and B⇀.
The direction of A⇀ × B ⇀ is perpendicular to the plane formed by A⇀ and B⇀, and obeys the right hand rule:
Position the middle and index fingers and thumb of your right hand at right angles with the index finger pointing straight. If the middle and index fingers approximate the directions of vectors A⇀ and B ⇀ respectively, then the thumb will be in the direction of A⇀ × B ⇀.
The components of cross product C⇀ are expressed in terms of those for A⇀ and B⇀ as follows:
Cx = Ay⋅Bz− Az⋅By
Cy=Az ⋅Bx − Ax⋅Bz
Cz = Ax⋅By− Ay⋅Bx
More compactly, the cross product can be written using a determinant:
A⇀ × B ⇀ = i⏞j⏞k⏞AxAyAzBxByBz
where, i⏞, j⏞, and k⏞ are unit vectors in the x, y, and z directions respectively.
Note: A⇀ × B ⇀ = −B⇀ × A⇀ (that is, the vector cross products are anticommutative).
Choose the coordinates of vectors A⇀ and B ⇀ and notice how the cross products change.
Vector A
Vector B
xA =
xB=
yA=
yB=
zA=
zB=
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