# Formulas for Vectors

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. These are the angles between the vector and the positive x-, y- and z- axes respectively of a rectangular system. The cosine of these angles, for vector $$\vec A$$ ,are given by:
$\cos (\alpha) = \dfrac{A_x}{|\vec A|} = \dfrac{A_x}{\sqrt{A_x^2 + A_y^2 + A_z^2}}$ $\cos (\beta) = \dfrac{A_y}{|\vec A|} = \dfrac{A_y}{\sqrt{A_x^2 + A_y^2 + A_z^2}}$ $\cos (\gamma) = \dfrac{A_z}{|\vec A|} = \dfrac{A_z}{\sqrt{A_x^2 + A_y^2 + A_z^2}}$

In 2-D, the direction of a vector is defined as an angle ( angle $$\theta$$ in the figure below ) that a vector makes with the positive x-axis. Figure 2. - Angle of a 2-D vector. 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

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. ### Geometrical Meaning of Cross Product

The area of a parallelogram defined by vectors $$\vec A$$ and $$\vec B$$ is the magnitude of their cross product given by: $\text{Area of Parallelogram} = |\vec A \times \vec B| = |\vec A | \cdot |\vec B| \cdot |\sin \theta|$