Define cross product of two vectors. If $$\vec{a} = \vec{i} + 3\vec{j} + 4\vec{k}$$  and $$\vec{b} = 2\vec{i} + 7\vec{j} – 5\vec{k}$$ find the vector $$\vec{b} \times \vec{a} \enspace and \enspace \vec{a} \times \vec{b}$$

Let $$\vec{a}$$ and $$\vec{b}$$ are two non vectors. Then the cross product of $$\vec{a}$$ and $$\vec{b}$$ is denoted by $$\vec{b} \times \vec{a}$$ and is defined as

$$\vec{a} \times \vec{b} = |\vec{a}| \enspace |\vec{b}| sin\theta$$

where $$\theta$$ be the angle between $$\vec{a}$$ and $$\vec{b}$$ and $$\hat{n}$$ is unit vector along $$( \vec{a} \times \vec{b} )$$

Problem Part:

Let

$$\vec{a} = \vec{i} + 3\vec{j} + 4\vec{k}$$  and $$\vec{b} = 2\vec{i} + 7\vec{j} – 5\vec{k}$$

Then

$$\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k}\\ 1 & 3 & 4\\ 2 & 7 & 5 \end{vmatrix}$$

$$= (-15-28)\vec{i} – (-5-8)\vec{j} + (7-6)\vec{k}$$

$$= (-43)\vec{i} + (13)\vec{j} + \vec{k}$$

and we know $$\vec{a} \times \vec{b} \enspace =\enspace – \vec{b} \times \vec{a}$$

so

$$\vec{b} \times \vec{a} \enspace =\enspace – \vec{a} \times \vec{b}$$

$$= 43\vec{i} – 13\vec{j} – \vec{k}$$