Vector Product of two vectors in Algebra (Cross Product)

The vector product of two non-null and non-parallel vectors a and b is expressed as:

a × b = |a||b| sinθ n̂ = ab sinθ n̂

The unit vector n̂ along a × b is given by:

\[\hat{\mathbf{n}}=\frac{\mathbf{a}\times\mathbf{b}}{|\mathbf{a}\times\mathbf{b}|}\]

\[\sin\theta=\frac{\left|\overline{a}\times\overline{b}\right|}{\left|\overline{a}\right|\left|\overline{b}\right|}\]

1. Determinant form:

If \[\overline{\mathrm{a}}=\mathrm{a}_{1}\hat{\mathrm{i}}+\mathrm{a}_{2}\hat{\mathrm{j}}+\mathrm{a}_{3}\hat{\mathrm{k}}\] and \[\overline{\mathrm{b}}=\mathrm{b}_1\hat{\mathrm{i}}+\mathrm{b}_2\hat{\mathrm{j}}+\mathrm{b}_3\hat{\mathrm{k}}\], then

\[\overline{\mathrm{a}}\times\overline{\mathrm{b}}= \begin{vmatrix} \hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ \mathbf{a}_1 & \mathbf{a}_2 & \mathbf{a}_3 \\ \mathbf{b}_1 & \mathbf{b}_2 & \mathbf{b}_3 \end{vmatrix}\]

2. Condition for zero cross product:

a × b = 0 ⇒ vectors are parallel (or one is zero)