Overview on scalar dot product and vector cross product with examples

Formula:

$$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$$

In terms of components, if $\vec{a} = \langle a_1, a_2, a_3 \rangle$ and $\vec{b} = \langle b_1, b_2, b_3 \rangle$, then:

$$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$$

Example:
Let $\vec{a} = \langle 1, 2, 3 \rangle$ and $\vec{b} = \langle 4, 5, 6 \rangle$.

$$\vec{a} \cdot \vec{b} = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32$$

The vector cross product of two vectors $\vec{a}$ and $\vec{b}$ is a vector quantity that is perpendicular to both $\vec{a}$ and $\vec{b}$. Its magnitude is the product of their magnitudes and the sine of the angle between them.

Formula:

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

where $\hat{n}$ is a unit vector perpendicular to the plane containing $\vec{a}$ and $\vec{b}$, determined by the right-hand rule.

In terms of components, if $\vec{a} = \langle a_1, a_2, a_3 \rangle$ and $\vec{b} = \langle b_1, b_2, b_3 \rangle$, then:

$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = (a_2 b_3 - a_3 b_2) \hat{i} - (a_1 b_3 - a_3 b_1) \hat{j} + (a_1 b_2 - a_2 b_1) \hat{k}$$

Example:
Let $\vec{a} = \langle 1, 2, 3 \rangle$ and $\vec{b} = \langle 4, 5, 6 \rangle$.

$$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{vmatrix}$$ $$= (2 \cdot 6 - 3 \cdot 5) \hat{i} - (1 \cdot 6 - 3 \cdot 4) \hat{j} + (1 \cdot 5 - 2 \cdot 4) \hat{k}$$ $$= (12 - 15) \hat{i} - (6 - 12) \hat{j} + (5 - 8) \hat{k}$$ $$= -3 \hat{i} + 6 \hat{j} - 3 \hat{k} = \langle -3, 6, -3 \rangle$$