Cross Product in 3D (A × B)
The cross product is a vector operation defined for 3D vectors. Given two vectors
\(A=\langle A_x,A_y,A_z\rangle\) and \(B=\langle B_x,B_y,B_z\rangle\), the cross product \(A\times B\)
produces a new vector that is perpendicular to the plane containing \(A\) and \(B\).
This makes the cross product a powerful tool for describing orientation and rotational effects
in geometry and physics.
Component-wise, the cross product is computed by:
\[
A\times B =
\left\langle
A_yB_z - A_zB_y,\;
A_zB_x - A_xB_z,\;
A_xB_y - A_yB_x
\right\rangle.
\]
You can also view it as a determinant:
\[
A\times B =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
A_x & A_y & A_z \\
B_x & B_y & B_z
\end{vmatrix},
\]
where \(\mathbf{i},\mathbf{j},\mathbf{k}\) are unit vectors along the \(x,y,z\) axes.
The magnitude \(\lVert A\times B\rVert\) has a direct geometric meaning:
\[
\lVert A\times B\rVert = \lVert A\rVert\,\lVert B\rVert\,\sin\theta,
\]
where \(\theta\) is the angle between \(A\) and \(B\) (from \(0\) to \(\pi\)). This magnitude equals the
area of the parallelogram spanned by the two vectors. Therefore, the area of the triangle
formed by \(A\) and \(B\) is \(\frac12\lVert A\times B\rVert\). If \(A\) and \(B\) are parallel (or if either vector is
the zero vector), then \(\sin\theta=0\) and the cross product becomes the zero vector.
The direction of \(A\times B\) is determined by the right-hand rule:
point your index finger along \(A\), your middle finger along \(B\), and your thumb points in the direction of \(A\times B\).
Swapping the order reverses the direction, because \(A\times B = -(B\times A)\). This “order matters” property is essential
in applications where orientation is important, such as computing normals in 3D graphics or determining rotation directions.
A common physics application is torque:
\(\tau = r\times F\), where \(r\) is a position vector (lever arm) and \(F\) is force.
The magnitude \(\lVert\tau\rVert = \lVert r\rVert\lVert F\rVert\sin\theta\) measures how strongly the force tends to rotate an object,
while the direction indicates the axis of rotation. Another example is angular momentum,
\(L = r\times p\), where \(p\) is momentum.
This calculator computes \(A\times B\), \(\lVert A\times B\rVert\), the angle \(\theta\), and the areas derived from the magnitude.
The interactive 3D view helps you see the plane spanned by \(A\) and \(B\), and the perpendicular direction of the cross product vector,
reinforcing both the geometry (area) and the orientation (right-hand rule).
