A three-dimensional vector is a quantity that has both magnitude and direction in three-dimensional space. Unlike two-dimensional vectors that exist only on a plane (x- and -axes), three-dimensional vectors exist in space with three coordinate axes (, , and axes).
A vector in three-dimensional space has three scalar coordinate components. Multiplying each scalar by its axis unit vector gives the corresponding vector projection:
a=(ax,ay,az)=axi+ayj+azk
The components in this form are:
ax is the scalar component along the x-axis, while axi is its vector projection
ay is the scalar component along the y-axis, while ayj is its vector projection
az is the scalar component along the z-axis, while azk is its vector projection
The cross product between two vectors a and b produces a vector whose dot product with each factor is zero:
(a×b)⋅a=(a×b)⋅b=0
a×b=(aybz−azby,azbx−axbz,axby−aybx)
The magnitude of the cross product is:
∣a×b∣=∣a∣∣b∣sinθ
In this formula, θ is the angle between the two vectors.
When a×b=0, its direction is perpendicular to both factors and follows the right-hand rule from a toward b. If either factor is zero or the factors are parallel, the cross product is 0 and therefore has no geometric direction.
