How to calculate for vector a minus b and also a dot b and also a minus b

$$\vec{a} - \vec{b} = (a_x - b_x)\vec{i} + (a_y - b_y)\vec{j} + (a_z - b_z)\vec{k}$$

For example, if $\vec{a} = 3\vec{i} + 2\vec{j} + \vec{k}$ and $\vec{b} = \vec{i} - 4\vec{j} + 5\vec{k}$:

$$\vec{a} - \vec{b} = (3 - 1)\vec{i} + (2 - (-4))\vec{j} + (1 - 5)\vec{k}$$ $$\vec{a} - \vec{b} = 2\vec{i} + 6\vec{j} - 4\vec{k}$$

The result is a new vector.

2. Dot Product: $\vec{a} \cdot \vec{b}$

The dot product (also called scalar product) results in a scalar (a single number), not a vector. You multiply corresponding components and then sum the results.

$$\vec{a} \cdot \vec{b} = (a_x)(b_x) + (a_y)(b_y) + (a_z)(b_z)$$

For example, if $\vec{a} = 3\vec{i} + 2\vec{j} + \vec{k}$ and $\vec{b} = \vec{i} - 4\vec{j} + 5\vec{k}$:

$$\vec{a} \cdot \vec{b} = (3)(1) + (2)(-4) + (1)(5)$$ $$= 3 - 8 + 5$$ $$= 0$$

The result is a scalar: $\boxed{0}$.

3. Cross Product: $\vec{a} \times \vec{b}$

The cross product (also called vector product) results in a new vector that is perpendicular to both $\vec{a}$ and $\vec{b}$. It is calculated using a determinant.

$$\vec{a} \times \vec{b} = \vec{i}(a_y b_z - a_z b_y) - \vec{j}(a_x b_z - a_z b_x) + \vec{k}(a_x b_y - a_y b_x)$$

For example, if $\vec{a} = 3\vec{i} + 2\vec{j} + \vec{k}$ and $\vec{b} = \vec{i} - 4\vec{j} + 5\vec{k}$:

$$\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ 3 & 2 & 1 \\ 1 & -4 & 5 \end{vmatrix}$$ $$= \vec{i}((2)(5) - (1)(-4)) - \vec{j}((3)(5) - (1)(1)) + \vec{k}((3)(-4) - (2)(1))$$ $$= \vec{i}(10 - (-4)) - \vec{j}(15 - 1) + \vec{k}(-12 - 2)$$ $$= \vec{i}(14) - \vec{j}(14) + \vec{k}(-14)$$ $$= 14\vec{i} - 14\vec{j} - 14\vec{k}$$

The result is a new vector: $\boxed{14\vec{i} - 14\vec{j} - 14\vec{k}}$.