Cross Product of Vector

Cross Product of Vector

Let \(\overrightarrow{a}\), \(\overrightarrow{b}\) be two vectors. The cross product or vector product or skew product of vectors \(\overrightarrow{a}\), \(\overrightarrow{b}\) is denoted by \(\overrightarrow{a}\times \overrightarrow{b}\)  and  is defined as follows.

i) If \(~\overrightarrow{a}\,=\,0\) or \(\overrightarrow{b}\,=\,0\) or \(\overrightarrow{a}\),\(\overrightarrow{b}\) are parallel then \(\overrightarrow{a}\times \overrightarrow{b}=0\).

ii) If  \(~\overrightarrow{a}\,\ne \,0\), \(~\overrightarrow{b}\,\ne \,0\), \(\overrightarrow{a}\),\(\overrightarrow{b}\) are not  parallel then \(\overrightarrow{a}\times \overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|(\sin \theta )\widehat{n}\) where \(\widehat{n}\) is a unit vector perpendicular to  \(\overrightarrow{a}\)  and \(\overrightarrow{b}\) so that \(\overrightarrow{a}\), \(\overrightarrow{b}\), \(\widehat{n}\) form a right handed system.

Note:

i)  \(\overrightarrow{a}\times \overrightarrow{b}\) is a vector.

ii) if \(\overrightarrow{a}\), \(\overrightarrow{b}\) are not parallel then \(\overrightarrow{a}\times \overrightarrow{b}\) is perpendicular to both \(\overrightarrow{a}\) and \(\overrightarrow{b}\).

iii) If \(\overrightarrow{a}\), \(\overrightarrow{b}\) are not parallel then \(\overrightarrow{a}\), \(\overrightarrow{b}\), \(\overrightarrow{a}\times \overrightarrow{b}\) from a right handed system.

iv) If \(\overrightarrow{a}\), \(\overrightarrow{b}\) are not parallel then \(|\overrightarrow{a}\times \overrightarrow{b}|=|\overrightarrow{a}||\overrightarrow{b}|\sin (\overrightarrow{a},\overrightarrow{b})\) and hence \(|\overrightarrow{a}\times \overrightarrow{b}|\le |\overrightarrow{a}||\overrightarrow{b}|\).

v) For any vector \(\overrightarrow{a}.\overrightarrow{a}\times \overrightarrow{b}=0\).

Example: If a = 2i – 3j +k and b = i + 4j – 2k then find (a + b) x (a – b)

Solution: a + b = 2i – 3j +k + i + 4j – 2k

= 3i + j – k

a – b = 2i – 3j + k – (i + 4j – 2k)

= i – 7j + 3k

\(\left( a+b \right)\times \left( a-b \right)=\left|\begin{matrix}   i & j & k  \\   3& 1 & -1  \\   1 & -7 & 3  \\\end{matrix} \right|=0\).

= i (3 – 7) – j (9 + 1) + k (-21 – 1)

= -4i – 10j – 22k.