Analytic Geometry

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Vector Product

Vectors: \(\mathbf{u},\) \(\mathbf{v},\) \(\mathbf{w}\)
Vector lengths: \(\left| \mathbf{u} \right|,\) \(\left| \mathbf{v} \right|,\) \(\left| \mathbf{w} \right|\)
Zero 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: \(\lambda,\) \(\mu\)
Area of parallelogram: \(S\)
  1. The vector product (also called the cross product) of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is the third vector \(\mathbf{w}\) whose length is equal to the product of their lengths and the sine of the angle between them, and which is perpendicular to them. The direction of the vector product is chosen so that the three vectors \(\mathbf{u},\) \(\mathbf{v},\) \(\mathbf{w}\) form a right handed system: \(\mathbf{u} \times \mathbf{v} = \mathbf{w},\) where
    \(\left| \mathbf{w} \right| = \left| \mathbf{u} \right| \cdot \left| \mathbf{v} \right| \cdot \sin \theta,\;\) \(0 \le \theta \le {\large\frac{\pi }{2}\normalsize};\)
    \(\mathbf{w} \bot \mathbf{u},\;\) \(\mathbf{w} \bot \mathbf{v};\)
    \(\mathbf{u}\), \(\mathbf{v}\), \(\mathbf{w}\) form a right-handed screw.
  2. The vector (cross) product of two vectors
  3. Vector 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
    \(\mathbf{w} = \mathbf{u} \times \mathbf{v} =\) \( \left| {\begin{array}{*{20}{c}} \mathbf{i} & \mathbf{j} & \mathbf{k}\\ {{X_1}} & {{Y_1}} & {{Z_1}}\\ {{X_2}} & {{Y_2}} & {{Z_2}} \end{array}} \right| =\) \( \left| {\begin{array}{*{20}{c}} {{Y_1}} & {{Z_1}}\\ {{Y_2}} & {{Z_2}} \end{array}} \right|\mathbf{i} \) \(-\, \left| {\begin{array}{*{20}{c}} {{X_1}} & {{Z_1}}\\ {{X_2}} & {{Z_2}} \end{array}} \right|\mathbf{j} \) \(+\, \left| {\begin{array}{*{20}{c}} {{X_1}} & {{Y_1}}\\ {{X_2}} & {{Y_2}} \end{array}} \right|\mathbf{k}.\)
  4. The absolute value of the vector product of \(\mathbf{u}\) and \(\mathbf{v}\) is equal to the area of the parallelogram spanned by these vectors:
    \(S = \left| {\mathbf{u} \times \mathbf{v}} \right| =\) \( \left| \mathbf{u} \right| \cdot \left| \mathbf{v} \right| \cdot \sin \theta \)
  5. Angle between two vectors expressed through their vector product
    \(\sin \theta = {\large\frac{{\left| {\mathbf{u} \times \mathbf{v}} \right|}}{{\left| \mathbf{u} \right| \cdot \left| \mathbf{v} \right|}}\normalsize}\)
  6. Noncommutative property of the vector product
    \(\mathbf{u} \times \mathbf{v} =\) \( – \left( {\mathbf{v} \times \mathbf{u}} \right)\)
  7. Associative property of the vector product over multiplication by a number
    \(\left( {\lambda \mathbf{u}} \right) \times \left( {\mu \mathbf{v}} \right) =\) \( \lambda \mu \mathbf{u} \times \mathbf{v}\)
  8. Distributive property of the vector product over addition of vectors
    \(\mathbf{u} \times \left( {\mathbf{v} + \mathbf{w}} \right) =\) \( \mathbf{u} \times \mathbf{v} \) \(+\; \mathbf{u} \times \mathbf{w}\)
  9. The vector product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is the zero vector if \(\mathbf{u}\) and \(\mathbf{v}\) are parallel (collinear):
    \(\mathbf{u} \times \mathbf{v} = \mathbf{0}\), if \(\mathbf{u}\parallel \mathbf{v} \left( {\theta = 0} \right)\).
  10. Vector product of the unit vectors
    \(\mathbf{i} \times \mathbf{i} = \mathbf{j} \times \mathbf{j} =\) \( \mathbf{k} \times \mathbf{k} \) \(= \mathbf{0}\)
  11. Vector product of distinct unit vectors
    \(\mathbf{i} \times \mathbf{j} = \mathbf{k},\;\) \(\mathbf{j} \times \mathbf{k} = \mathbf{i},\;\) \(\mathbf{k} \times \mathbf{i} = \mathbf{j}.\)