# Formulas

## Analytic Geometry

Vectors: $$\mathbf{u},$$ $$\mathbf{v},$$ $$\mathbf{w},$$ $$\mathbf{u_1},$$ $$\mathbf{u_2},\;\ldots\;$$
Null vector: $$\mathbf{0}$$
Coordinates of vectors: $${X_1},$$ $${Y_1},$$ $${Z_1},$$ $${X_2},$$ $${Y_2},$$ $${Z_2}$$
1. The sum of two vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ is the third vector $$\mathbf{w}$$ drawn from the tail of $$\mathbf{u}$$ to the head of $$\mathbf{v}$$ if the tail of the vector $$\mathbf{v}$$ is placed at the head of $$\mathbf{u}$$. Vector addition is performed using the triangle or parallelogram rule.
$$\mathbf{w} = \mathbf{u} + \mathbf{v}$$
2. The sum of several vectors $$\mathbf{u_1},$$ $$\mathbf{u_2},$$ $$\mathbf{u_3}, \ldots$$ is called the vector $$\mathbf{w}$$ resulting from the sequential addition of these vectors. This operation is performed by the polygon rule.
$$\mathbf{w} = \mathbf{u_1} + \mathbf{u_2} + \mathbf{u_3} + \ldots$$ $$+\; \mathbf{u_n}$$
$$\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$$
$$\left( {\mathbf{u} + \mathbf{v}} \right) + \mathbf{w} =$$ $$\mathbf{u} + \left( {\mathbf{v} + \mathbf{w}} \right)$$
5. Sum of vectors in coordinate form
$$\mathbf{u} + \mathbf{v} =$$ $$\big( {{X_1} + {X_2}, }$$ $${{Y_1} + {Y_2}, }$$ $${{Z_1} + {Z_2}} \big)$$
6. The difference of two vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ is called the vector $$\mathbf{w}$$ provided that
$$\mathbf{w} = \mathbf{u} – \mathbf{v},$$ if $$\mathbf{w} + \mathbf{v} = \mathbf{u}$$
7. The difference of the vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ is equal to the sum of $$\mathbf{u}$$ and the opposite vector $$-\mathbf{v}$$:
$$\mathbf{u} – \mathbf{v} =$$ $$\mathbf{u} + \left( -\mathbf{v} \right)$$
$$\mathbf{u} – \mathbf{u} = \mathbf{0}$$
$$\left| \mathbf{0} \right| = 0$$
$$\mathbf{u} – \mathbf{v} = \big( {{X_1} – {X_2}, }$$ $${{Y_1} – {Y_2}, }$$ $${{Z_1} – {Z_2}} \big)$$