# Formulas and Tables

Analytic Geometry# Vector Coordinates

Vector lengths: \(\left| {\mathbf{r}} \right| ,\) \(\left| {\mathbf{AB}} \right| \)

Unit vectors: \(\mathbf{i},\) \(\mathbf{j},\) \(\mathbf{k}\)

Coordinates of points: \({x_0},\) \({y_0},\) \({z_0},\) \({x_1},\) \({y_1},\) \({z_1}\)

Direction cosines: \(\cos \alpha,\) \(\cos \beta,\) \(\cos \gamma\)

- A vector is a directed line segment, one end of which is the beginning and the other is the end of the vector.
- 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. - Any vector can be expanded as a linear combination of the basis vectors. The decomposition formula is written in the form

\(\mathbf{r} = \mathbf{AB} =\) \(\left( {{x_1} – {x_0}} \right)\mathbf{i} \) \(+\left( {{y_1} – {y_0}} \right)\mathbf{j} \) \(+\left( {{z_1} – {z_0}} \right)\mathbf{k}.\)

- The length (or magnitude) of a vector is the distance between the initial point and end point of the vector:

\(\left| \mathbf{r} \right| = \left| \mathbf{AB} \right| \) \(= \Big[ {{{\left( {{x_1} – {x_0}} \right)}^2} }\) \(+\;{ {{\left( {{y_1} – {y_0}} \right)}^2} }\) \(+\;{ {{\left( {{z_1} – {z_0}} \right)}^2}}\Big]^{\large{\frac{1}{2}}\normalsize}. \) - Opposite vectors have the same lengths but opposite direction:

If \(\mathbf{AB} = \mathbf{r}\), then \(\mathbf{BA} = -\mathbf{r}\).

- 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 \(\mathbf{r}\).

- Vectors are called collinear if they are parallel to the same straight line.
- 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 \(X = {X_1},\;\) \(Y = {Y_1},\;\) \(Z = {Z_1}\).