# 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.