Formulas and Tables

Analytic Geometry

Vector Addition and Subtraction

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.
    \(\mathbf{w} = \mathbf{u} + \mathbf{v}\)
The sum of two vectors using the triangle or parallelogram rule
  1. 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.
    \(\mathbf{w} = \mathbf{u_1} + \mathbf{u_2} + \mathbf{u_3} + \ldots\) \(+\; \mathbf{u_n}\)
The sum of several vectors
  1. Commutative law of addition
    \(\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\)
  2. ssociative law of addition
    \(\left( {\mathbf{u} + \mathbf{v}} \right) + \mathbf{w} =\) \( \mathbf{u} + \left( {\mathbf{v} + \mathbf{w}} \right)\)
  3. Sum of vectors in coordinate form
    When adding two vectors, the corresponding coordinates are added.
    \(\mathbf{u} + \mathbf{v} =\) \( \big( {{X_1} + {X_2}, }\) \({{Y_1} + {Y_2}, }\) \({{Z_1} + {Z_2}} \big)\)
  4. 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 \(\mathbf{w} + \mathbf{v} = \mathbf{u}\)
The difference of two vectors
  1. The difference of the vectors \(\mathbf{u}\) and \(\mathbf{v}\) is equal to the sum of \(\mathbf{u}\) and the opposite vector \(-\mathbf{v}\):
    \(\mathbf{u} – \mathbf{v} =\) \( \mathbf{u} + \left( -\mathbf{v} \right) \)
  2. The difference of two equal vectors is the null vector:
    \(\mathbf{u} – \mathbf{u} = \mathbf{0} \)
  3. The length of the null vector is zero:
    \(\left| \mathbf{0} \right| = 0\)
  4. Difference of vectors in coordinate form
    When subtracting two vectors, the corresponding coordinates are subtracted:
    \(\mathbf{u} – \mathbf{v} = \big( {{X_1} – {X_2}, }\) \({{Y_1} – {Y_2}, }\) \({{Z_1} – {Z_2}} \big)\)