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Vectors: magnitude, direction and products

Calculate and check a Cartesian cross product · Diagrams

Cyclic products in a right-handed basis

Use a right-handed orthonormal basis. Every cross-product result is a vector.

i:
+x
j:
+y
k:
+z
Vector results · row first, column second
×ijk
i

0\mathbf0

k\mathbf k

−j-\mathbf j

j

−k-\mathbf k

0\mathbf0

i\mathbf i

k

j\mathbf j

−i-\mathbf i

0\mathbf0

Follow cyclic order

i×j=kj×k=ik×i=j\begin{aligned}\mathbf i\times\mathbf j&=\mathbf k\\\mathbf j\times\mathbf k&=\mathbf i\\\mathbf k\times\mathbf i&=\mathbf j\end{aligned}

Take two consecutive directions in the order i, j, k, then wrap back to i. The result is the remaining positive basis direction. The arrow diagram records this order; the original axes show the spatial directions.

Reverse the order to negate the vector

a×b=−(b×a)\mathbf a\times\mathbf b=-(\mathbf b\times\mathbf a)

Reversing a cyclic pair gives the opposite basis direction. The diagonal products are the zero vector because each direction is parallel to itself; zero has no unit direction.

Compare with the spatial source sketch

Expand diagram

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Three spatially perpendicular axes x,y,z carry i,j,k. Positive cyclic products i×j=k, j×k=i, k×i=j; reversing each order negates its result. Self-cross-products are zero. Screen projection is schematic.
Three spatially perpendicular axes x,y,z carry i,j,k. Positive cyclic products i×j=k, j×k=i, k×i=j; reversing each order negates its result. Self-cross-products are zero. Screen projection is schematic.
Three spatially perpendicular axes x,y,z carry i,j,k. Positive cyclic products i×j=k, j×k=i, k×i=j; reversing each order negates its result. Self-cross-products are zero. Screen projection is schematic.
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