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# Formulas and Tables

Analytic Geometry

# Dot Product

Vectors: $$\mathbf{u},$$ $$\mathbf{v},$$ $$\mathbf{w}$$
Vector lengths: $$\left| \mathbf{u} \right|,$$ $$\left| \mathbf{v} \right|$$
Null vector: $$\mathbf{0}$$
Unit vectors: $$\mathbf{i},$$ $$\mathbf{j},$$ $$\mathbf{k}$$

Angle between two vectors: $$\theta$$
Coordinates of vectors: $${X_1},$$ $${Y_1},$$ $${Z_1},$$ $${X_2},$$ $${Y_2},$$ $${Z_2}$$
Real numbers: $$\lambda,$$ $$\mu$$

1. The dot product (also called the scalar product) of two vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ is the product of their lengths and the cosine of the angle between them.
$$\mathbf{u} \cdot \mathbf{v} = \left| \mathbf{u} \right|\left| \mathbf{v} \right|\cos \theta$$
1. Dot product in coordinate form
If $$\mathbf{u} = \left( {{X_1},{Y_1},{Z_1}} \right)$$, $$\mathbf{v} = \left( {{X_2},{Y_2},{Z_2}} \right),$$ then
$$\mathbf{u} \cdot \mathbf{v} = {X_1}{X_2}$$ $$+\; {Y_1}{Y_2}$$ $$+\; {Z_1}{Z_2}$$.
2. Angle between two vectors
If $$\mathbf{u} = \left( {{X_1},{Y_1},{Z_1}} \right)$$, $$\mathbf{v} = \left( {{X_2},{Y_2},{Z_2}} \right)$$, then
$$\cos \theta = {\large\frac{{\mathbf{u} \cdot \mathbf{v}}}{{\left| \mathbf{u} \right| \cdot \left| \mathbf{v} \right|}}\normalsize} =$$ $${\large\frac{{{X_1}{X_2} + {Y_1}{Y_2} + {Z_1}{Z_2}}}{{\sqrt {X_1^2 + Y_1^2 + Z_1^2} \sqrt {X_2^2 + Y_2^2 + Z_2^2} }}\normalsize}.$$
It is assumed here that $$\mathbf{u}$$ and $$\mathbf{v}$$ are non-zero vectors.
3. Commutative law for dot product
$$\mathbf{u} \cdot \mathbf{u} = \mathbf{v} \cdot \mathbf{u}$$
4. Associative law for dot product
$$\left( {\lambda \mathbf{u}} \right) \cdot \left( {\mu \mathbf{v}} \right) =$$ $$\lambda \mu \mathbf{u} \cdot \mathbf{v}$$
5. Distributive law for dot product
$$\mathbf{u} \cdot \left( {\mathbf{v} + \mathbf{w}} \right) =$$ $$\mathbf{u} \cdot \mathbf{v}$$ $$+\; \mathbf{u} \cdot \mathbf{w}$$
6. The dot product of the vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ is zero if the vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ are perpendicular, or if one of the vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ or both of them are zero.
$$\mathbf{u} \cdot \mathbf{v} = 0,$$ if $$\mathbf{u} \bot \mathbf{v}\left( {\theta = \large\frac{\pi }{2}}\normalsize \right)$$ or $$\mathbf{u} = \mathbf{0}$$ and/or $$\mathbf{v} = \mathbf{0}$$.
7. The dot product of two vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ is positive if the angle $$\theta$$ between the vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ is acute.
$$\mathbf{u} \cdot \mathbf{v} \gt 0$$, if $$0 \lt \theta \lt {\large\frac{\pi }{2}\normalsize}$$.
8. The dot product of two vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ is negative if the angle $$\theta$$ between the vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ is obtuse.
$$\mathbf{u} \cdot \mathbf{v} \lt 0$$, if $${\large\frac{\pi }{2}\normalsize} \lt \theta \lt \pi$$.
9. The dot product of two vectors is less than or equal to the product of their lengths:
$$\mathbf{u} \cdot \mathbf{v} \le \left| \mathbf{u} \right| \cdot \left| \mathbf{v} \right|$$
10. The dot product of two vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ is equal to the product of their lengths if only the vectors $$\mathbf{u}$$ and $$\mathbf{v}$$ are parallel:
$$\mathbf{u} \cdot \mathbf{v} = \left| \mathbf{u} \right| \cdot \left| \mathbf{v} \right|,$$ if $$\mathbf{u}\parallel \mathbf{v}\left( {\theta = 0} \right)$$.
11. The scalar square of a vector is equal to the square of its length:
If $$\mathbf{u} = \left( {{X_1},{Y_1},{Z_1}} \right),$$ then $$\mathbf{u} \cdot \mathbf{u} = {{\mathbf{u}}^2} =$$ $${\left| \mathbf{u} \right|^2} =$$ $$X_1^2 + Y_1^2 + Z_1^2.$$
12. Scalar squares of the unit vectors
$$\mathbf{i} \cdot \mathbf{i} = \mathbf{j} \cdot \mathbf{j} =$$ $$\mathbf{k} \cdot \mathbf{k}$$ $$= 1$$
13. Dot product of distinct unit vectors
$$\mathbf{i} \cdot \mathbf{j} = \mathbf{j} \cdot \mathbf{k} =$$ $$\mathbf{k} \cdot \mathbf{i}$$ $$= 0$$