NotesAtHKU

Vector can be represented by $\vec{F}$ or $\boldsymbol{F}$ . The magnitude can be represented by $\|F\|$ or $F$ .

They are vectors with magnitude 1: $\hat{A}=\frac{\vec{A}}{\|\vec{A}\|}$

Relative vectors (resultant vectors)

We can find the direction in which another object moves relative to you (your movement vector) by the following steps:

Draw the two vectors pointing head-to-head
Draw the resultant vector from the tail of the other vector to your vector

Coplanar vectors

Cartesian vector notation

In two dimensions, the Cartesian unit vectors $\boldsymbol{i}, \boldsymbol{j}$ are used to designate the directions of the x and y axes respectively.

$F=F_x \boldsymbol{i} + F_y \boldsymbol{j}$

Where $F_{x/y}$ is the $x/y$ component of $F$ . And to find the x/y components we can use trigonometry:

$F_x=F\cos{\theta}, F_y=F\sin{\theta}$

Where the angle $\theta$ is the angle between $F$ and the x-axis.

Resultant force

The resultant force $F_R$ can be found by the sum of the components of $F$ :

$F_R=\sum F$

In Cartesian form, it's the same as adding all the terms together: $F_R=(F_{x1}+F_{x2})\boldsymbol{i}+(F_{y1}+F_{y2})\boldsymbol{j}$

Orientation of vector

We always consider the angle between $F$ & $F_x$ . It can be found by $\theta=\tan^{-1}\frac{F_y}{F_x}$ .

Magnitude of forces

The magnitude will simply be the square root of the sum of squared components of the force:

$\|F\|=\sqrt{F_x^2+F_y^2+\dots}$

Converting vectors to Cartesian form

Given a vector $F$ with magnitude $\|F\|$ and angle $\theta$ :

$\vec{F}=\|F\|\times\hat{F}=\|F\|\times\frac{r}{\|r\|}$

Where $r$ is the position vector of the point.

Vectors in 3D

The concepts above can be extended to 3D simply by adding another variable to the system.

Coordinate direction angles

The direction of A is defined by the coordinate direction angles: $\alpha, \beta, \gamma$ , which are measured between the tail of A and the positive $x, y, z$ axes.

A\cos\alpha=A_x,\quad A\cos\beta=A_y,\quad A\cos\gamma=A_z

Vectors