AcademyVectors In Rn
Academy
Vector Product
Level 1 - Linear Algebra topic page in Vectors In Rn.
Principle
The vector product, also called the cross product, is defined for vectors in \(\mathbb R^3\). It takes two vectors and returns a new vector perpendicular to both. Its length measures the area of the parallelogram spanned by the two input vectors.
Notation
\(\mathbf u\times\mathbf v\)
the vector product of \mathbf u and \mathbf v
\(\mathbf i,\mathbf j,\mathbf k\)
standard coordinate vectors in \mathbb R^3
\(\theta\)
the angle between two non-zero vectors
\(|\mathbf u\times\mathbf v|\)
area of the parallelogram spanned by \mathbf u and \mathbf v
\(\mathbf 0\)
the zero vector
The Core Method
For \(\mathbf u=(u_1,u_2,u_3)\) and \(\mathbf v=(v_1,v_2,v_3)\), compute
Cross product
\[\mathbf u\times\mathbf v=(u_2v_3-u_3v_2,\;u_3v_1-u_1v_3,\;u_1v_2-u_2v_1)\]
The result is perpendicular to both inputs:
Perpendicular output
\[(\mathbf u\times\mathbf v)\cdot\mathbf u=0,\qquad (\mathbf u\times\mathbf v)\cdot\mathbf v=0\]
Its length is
Cross product magnitude
\[|\mathbf u\times\mathbf v|=|\mathbf u|\,|\mathbf v|\sin\theta\]
The order matters:
Anti-commutativity
\[\mathbf u\times\mathbf v=-(\mathbf v\times\mathbf u)\]
Use the right-hand rule to choose the positive direction.
Worked Cases
Question
Compute \((1,2,3)\times(4,-1,2)\).
Answer
Use the component formula. The first component is \(2\cdot2-3\cdot(-1)=4+3=7\). The second component is \(3\cdot4-1\cdot2=12-2=10\). The third component is \(1\cdot(-1)-2\cdot4=-1-8=-9\). Therefore \((1,2,3)\times(4,-1,2)=(7,10,-9)\).
Examples
Question
What are \(\mathbf i\times\mathbf j\), \(\mathbf j\times\mathbf k\), and \(\mathbf k\times\mathbf i\)?
Answer
Using the standard right-handed coordinate system, \(\mathbf i\times\mathbf j=\mathbf k\), \(\mathbf j\times\mathbf k=\mathbf i\), and \(\mathbf k\times\mathbf i=\mathbf j\). Reversing the order changes the sign, so \(\mathbf j\times\mathbf i=-\mathbf k\).