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.
    \(\mathbf{u} \cdot \mathbf{v} = \left| \mathbf{u} \right|\left| \mathbf{v} \right|\cos \theta \)
The dot (scalar) product of two vectors
  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
    \(\mathbf{i} \cdot \mathbf{j} = \mathbf{j} \cdot \mathbf{k} =\) \( \mathbf{k} \cdot \mathbf{i} \) \(= 0\)