Here are some steps for Cross product

**Step 1 :** Read the problem and note down the given vectors of the form

$\vec{u}$ = a_{1} i + b_{1} j + c_{1} k

$\vec{v}$ = a_{2} i + b_{2} j + c_{2} k.

**Step 2 :** Write the matrix form Using given vectors and then calculate the determinant of it. Solve the determinant as below to get the answer

u $\times$ v = $\begin{vmatrix}

a_{1} & b_{1} & c_{1}\\

a_{2} & b_{2} & c_{2}\\

a_{3} & b_{3} & c_{3}

\end{vmatrix}$

= a_{1} (b_{2}c_{3} - b_{3} c_{2}) - b_{1 }(a_{2}c_{3} - a_{3}c_{2}) + c_{1} (a_{2}b_{3} - a_{3}b_{2})

Substituting the values in the determinant to get the answer.

Below are given some solved problems on cross product which may be helpful for you. ### Solved Examples

**Question 1: **Find the cross product of u = (3, 5, 7) and v = (-1, 4, 2).

** Solution: **

**Step 1: **The given vectors are

$\vec{u}$ = 3i + 5j + 7k

$\vec{v}$ = -i + 4j + 2k

Step 2: The cross product is given by

u $\times$ v = $\begin{vmatrix}
i & j & k\\
3 & 5 & 7\\
-1 & 4 & 2
\end{vmatrix}$

= i (5(2) - 4(7)) - j (3(2) - (-1)7) + k (3(4) - (-1)5)

= -18i - 13j + 17k.

**Question 2: **Find the cross product of u = (0, 5, 1) and v = (-1, 0, 2).

** Solution: **

Step 1: The given vectors are

$\vec{u}$ = 5j + k

$\vec{v}$ = -i + 2k

Step 2: The cross product is given by

u $\times$ v = $\begin{vmatrix}
i & j & k\\
0 & 5 & 1\\
-1 & 0 & 2
\end{vmatrix}$

= i (5(2) - 0(1)) - j (0(2) - (-1)1) + k (0(0) - (-1)5)

= 10 i - j + 5k.