Analytic Geometry

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Vector Coordinates

Vectors: \(\mathbf{r},\) \(\mathbf{r_1},\) \(\mathbf{AB}\)
Vector lengths: \(\left| {\mathbf{r}} \right| ,\) \(\left| {\mathbf{AB}} \right| \)
Unit vectors: \(\mathbf{i},\) \(\mathbf{j},\) \(\mathbf{k}\)
Vector coordinates: \(X,\) \(Y,\) \(Z,\) \({X_1},\) \({Y_1},\) \({Z_1}\)
Coordinates of points: \({x_0},\) \({y_0},\) \({z_0},\) \({x_1},\) \({y_1},\) \({z_1}\)
Direction cosines: \(\cos \alpha,\) \(\cos \beta,\) \(\cos \gamma\)
  1. A vector is a directed line segment, one end of which is the beginning and the other is the end of the vector.
  2. The unit vectors of the three-dimensional Cartesian coordinate system are denoted as follows:
    \(\mathbf{i} = \left( {1,0,0} \right),\;\) \(\mathbf{j} = \left( {0,1,0} \right),\;\) \(\mathbf{k} = \left( {0,0,1} \right)\),
    \(\left| \mathbf{i} \right| = \left| \mathbf{j} \right| = \left| \mathbf{k} \right| = 1\).
    This trio of unit vectors forms a basis of the coordinate system.
  3. Any vector can be expanded as a linear combination of the basis vectors. The decomposition formula is written in the form
    \(\mathbf{r} = \mathbf{AB} =\) \(\left( {{x_1} – {x_0}} \right)\mathbf{i} \) \(+\left( {{y_1} – {y_0}} \right)\mathbf{j} \) \(+\left( {{z_1} – {z_0}} \right)\mathbf{k}.\)
  4. Vector coordinates
  5. The length (or magnitude) of a vector is the distance between the initial point and end point of the vector:
    \(\left| \mathbf{r} \right| = \left| \mathbf{AB} \right| \) \(= \Big[ {{{\left( {{x_1} – {x_0}} \right)}^2} }\) \(+\;{ {{\left( {{y_1} – {y_0}} \right)}^2} }\) \(+\;{ {{\left( {{z_1} – {z_0}} \right)}^2}}\Big]^{\large{\frac{1}{2}}\normalsize}. \)
  6. Opposite vectors have the same lengths but opposite direction:
    If \(\mathbf{AB} = \mathbf{r}\), then \(\mathbf{BA} = -\mathbf{r}\).
  7. Opposite vectors
  8. The coordinates of a vector are defined as projections of the vector onto the coordinate axes:
    \(X = \left| \mathbf{r} \right|\cos \alpha,\;\) \(Y = \left| \mathbf{r} \right|\cos \beta,\;\) \(Z = \left| \mathbf{r} \right|\cos \gamma.\)
    The quantities \(\cos\alpha\), \(\cos\beta\), \(\cos\gamma\) are called direction cosines of the vector \(\mathbf{r}\).
  9. Direction cosines of the vector r
  10. Vectors are called collinear if they are parallel to the same straight line.
  11. Vectors are said to be equal (or equivalent) if they have the same direction and equal lengths. Equal vectors have equal coordinates.
    If \(\mathbf{r}\left( {X,Y,Z} \right) =\) \( \mathbf{r_1}\left( {{X_1},{Y_1},{Z_1}} \right)\), then \(X = {X_1},\;\) \(Y = {Y_1},\;\) \(Z = {Z_1}\).