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Analytic Geometry

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