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.