Some of the most important formulas for vectors such as the magnitude, the direction, the unit vector, addition, subtraction, scalar multiplication and cross product are presented.
Vector Defined by two Points
\( \) \( \)\( \) \( \) The components of a vector \( \vec {PQ} \) defined by two points \( P(P_x \;, \; P_y \;, \; P_z )\) (initial point) and \( Q(Q_x \;, \; Q_y \;, \; Q_z )\) (terminal point) are given as follows: \[ \vec{PQ} = \;< Q_x  P_x \;, \; Q_y  P_y \;, \; Q_z  P_z > \]
In what follows \( \vec A, \vec B \) and \( \vec C \) are 3 dimensional vectors given by their components as follows:
\( \vec A = \; < A_x \;, \; A_y \;, \; A_z > \)
\( \vec B = \; < B_x \;, \; B_y \;, \; B_z > \)
\( \vec C = \; < C_x \;, \; C_y \;, \; C_z > \)
Magnitude of a Vector
The magnitude of vector \( \vec A \) written as \( \vec A \) is given by
\[ \vec A = \sqrt{A_x^2 + A_y^2 + A_z^2} \]
Unit Vector
A unit vector is a vector whose magnitude is equal to 1.
The unit vector \( \vec u \) that has the same direction as vector \( \vec A \) is given by
\[ \vec u = \dfrac{\vec A}{\vec A} = \; < \dfrac{A_x}{\sqrt{A_x^2 + A_y^2 + A_z^2}} \;, \; \dfrac{A_y}{\sqrt{A_x^2 + A_y^2 + A_z^2}} \;,\; \dfrac{A_z}{\sqrt{A_x^2 + A_y^2 + A_z^2}} > \]
Direction of a Vector
In 3 dimensional space, the direction of a vector is defined by 3 angles \( \alpha \) , \( \beta \) and \( \gamma\) (see Figure 1. below) called direction cosines.
\[ \cos (\alpha) = \dfrac{A_x}{\vec A} = \dfrac{A_x}{\sqrt{A_x^2 + A_y^2 + A_z^2}} \] \[ \cos (\beta) = \dfrac{A_y}{\vec A} = \dfrac{A_y}{\sqrt{A_x^2 + A_y^2 + A_z^2}} \] \[ \cos (\gamma) = \dfrac{A_z}{\vec A} = \dfrac{A_z}{\sqrt{A_x^2 + A_y^2 + A_z^2}} \]
In 2D, the direction of a vector is defined as an angle ( angle \( \theta \) in the figure below ) that a vector makes with the positive xaxis.
Vector \( \vec A = \; < A_x \;, \; A_y > \) is given by \[ \theta = \arctan (\dfrac{A_y}{A_x}) \] taking into account the signs of \( A_x \) and \( A_y \) to determine the quadrant where the terminal side of the vector is located.
Operations on Vectors

Addition
The addition of vectors \( \vec A \) and \( \vec B \) is defined by \[ \vec A + \vec B = \; < A_x + B_x \; , \; A_y + B_y \; , \; A_z + B_z > \] More on Vector Addition. 
Subtraction
The subtraction of vectors \( \vec A \) and \( \vec B \) is defined by \[ \vec A  \vec B = \; < A_x  B_x \; , \; A_y  B_y \; , \; A_z  B_z > \] More on vector subtraction and adding and subtracting vectors.

Multiply Vector by a Scalar
The multiplication of a vector \( \vec A \) by a scalar \( k \) is defined by \[ k \vec A = \; < k A_x \; , \; k A_y \; , \; k A_z > \]
Scalar Product of Vectors
Definition
The Scalar (or dot) product of two vectors \( \vec A \) and \( \vec B \) is given by
\[ \vec A \cdot \vec B = \vec A \cdot \vec B \cdot\cos \theta \]
where \( \theta \) is the angle between vectors \( \vec A \) and \( \vec B \).
Given the coordinates of vectors \( \vec A \) and \( \vec B \), it can be shown that
\[ \vec A \cdot \vec B = A_x \cdot B_x + A_y \cdot B_y + A_z \cdot B_z \]
Properties of Scalar Product
\[ \vec A \cdot \vec B = \vec B \cdot \vec A \] \[ \vec A \cdot (\vec B + \vec C) = \vec A \cdot \vec B + \vec A \cdot \vec C \] \[ k \vec A \cdot (\vec B) = k (\vec A \cdot \vec B) \]
Orthogonal Vectors
Two vectors \( \vec A \) and \( \vec B \) are orthogonal, the angle \( \theta \) between equal to \( 90^{\circ} \), if and only if \[ \vec A \cdot \vec B = \vec A \cdot \vec B \cdot \cos \theta = \vec A \cdot \vec B \cdot \cos 90^{\circ} = 0\]
Angle Between Two Vectors
If \( \theta \) is the angle made by two vectors \( \vec A \) and \( \vec B \), then \[ \cos \theta = \dfrac{\vec A \cdot \vec B}{ \vec A\cdot \vec B} = \dfrac{A_x \cdot B_x + A_y \cdot B_y + A_z \cdot B_z}{ \sqrt{A_x^2 + A_y^2 + A_z^2} \cdot\sqrt{B_x^2 + B_y^2 + B_z^2}} \]
Cross Product
The cross product of two vectors \( \vec A \) and \( \vec B \) is a vector orthogonal to both vectors and is given by
\[ \vec A \times \vec B = \begin{vmatrix}
\vec i & \vec j & \vec k\\
A_x & A_y & A_z\\
B_x & B_y & B_z
\end{vmatrix} \\ = (A_y B_z  A_z B_y ) \vec i  (A_x B_z  A_z B_x) \vec j + (A_x B_y  A_y B_x) \vec k\]
Properties of Cross Product
\[ \vec A \times \vec B =  \vec B \times \vec A \] \[ (k \vec A) \times \vec B = \vec A \times (k \vec B ) = k( \vec A \times \vec B) \] The cross product is a vector and there may a need as in eletromagnetism and many other topics in physics to find the orientation of this vector. Use the right hand rule to find the orientation of the cross product: point the index in the direction of vector A, the middle finger in the direction of vector B and the direction of the cross product A × B is in the same direction of the thumb.