Answer
a) $\lt 6, -5,3 \gt$ or, $6i-5j+3k$
b) $\lt \dfrac{6}{\sqrt {70}},\dfrac{-5}{\sqrt {70}},\dfrac{3}{\sqrt {70}} \gt$
Work Step by Step
a) The cross product is defined as:
$u \times v=\begin{vmatrix}i&j&k\\m_1&m_2&m_3\\n_1&n_2&n_3\end{vmatrix}=\lt m_2n_3-m_3n_2, m_3n_1-m_1n_3, m_1n_2-m_2b_1 \gt$
Now, $u \times v=\lt (3)(2)-(5)(0),(5)(-1)-(0)(2), (0)(0) -(3)(-1) \gt=\lt 6, -5,3 \gt$ or, $6i-5j+3k$
b) The unit vector is defined as: $\hat{n}=\dfrac{u \times v}{|u \times v|}$
Here,$ |u \times v|=\sqrt {(6)^2+(-5)^2+(3)^2}=\sqrt {70}$
Now,
$\hat{n}=\dfrac{u \times v}{|u \times v|}=\dfrac{\lt 6, -5,3 \gt}{\sqrt {70}}=\lt \dfrac{6}{\sqrt {70}},\dfrac{-5}{\sqrt {70}},\dfrac{3}{\sqrt {70}} \gt$
