Cross Product - Maple Help

Cross Product

Main Concept

The cross product of $\stackrel{\mathbf{⇀}}{\mathbit{A}}$ and $\stackrel{\mathbf{⇀}}{\mathbit{B}}$  is a vector denoted $\stackrel{\mathbf{⇀}}{\mathbit{A}}$ × $\stackrel{\mathbf{⇀}}{\mathbit{B}}$. The magnitude of  is given by

 where θ is the angle between $\stackrel{⇀}{A}$ and $\stackrel{⇀}{B}$.

The direction of  is perpendicular to the plane formed by $\stackrel{⇀}{A}$ and $\stackrel{⇀}{B}$, 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 $\stackrel{\mathit{⇀}}{A}$ and  respectively, then the thumb will be in the direction of .

The components of cross product $\stackrel{⇀}{C}$ are expressed in terms of those for $\stackrel{⇀}{A}$ and $\stackrel{⇀}{B}$ as follows:

More compactly, the cross product can be written using a determinant:

where, $\stackrel{⏞}{\mathbf{i}}$, $\stackrel{⏞}{\mathbit{j}}$, and $\stackrel{⏞}{\mathbit{k}}$ are unit vectors in the $\mathbit{x}$, $\mathbit{y}$, and $\mathbit{z}$ directions respectively.

Note:        (that is, the vector cross products are anticommutative).

Choose the coordinates of vectors $\stackrel{\mathit{⇀}}{A}$ and  and notice how the cross products change.

 Vector A Vector B ${x}_{B}=$ ${y}_{A}=$ ${y}_{B}=$ ${z}_{A}=$ ${z}_{B}=$



 More MathApps