Wize University Statics Textbook (Master) > Force Vectors

Vectors and Their Components

Vectors and Their Components

A Vector is a quantity that has both magnitude and direction such as velocity.
A Scalar has magnitude but no direction such as speed.
Two vectors are equal to each other as long as they have the same size and direction
Vectors can be written in terms of their components based on different coordinate systems
The most common coordinate system is XYZ -coordinate system which is also known as Cartesian coordinate system.
Theses components are the shadow of the vector on different axis of the coordinate system
A given vector $\vec{a}$ in 2D space is shown as:

\boxed{\vec{a}=\left( a_x,a_y\right )=a_x\hat{i}+a_y\hat{j} }

- where vectors , $\hat{i}$ and $\hat{j}$ are basic unit vectors in $x$ and $y$ directions, respectively. These unit vectors are:

\begin{array}{l}\hat{i}=(1,0)\\\ \hat{j}=(0,1)\end{array}

Unit vectors are vectors with length equal to 1 which are used to show directions.
$a_x$ and $a_y$ are called components of vector $\vec{a}$

ٰVector's Magnitude

Magnitude of a 2D Vector

We can calculate the magnitude (a.k.a. length or size) of a vector in 2D using Pythagorean's Theorem:

Image illustrating Vector's Magnitude (2D & 3D)

Magnitude of a 3D Vector

Similarly, We can calculate the magnitude of a vector in 3D using an extension of this formula.

Components of a Vector Using the Angle It Makes Respect to x or y Axis

In many problems we need to find the components of a vector along different directions. For these problems we use trigonometric relationship using right triangle formed by the vector as the hypotenuse and components as the two other sides of the triangle.

Example:

Consider the following vector which makes an angle of $\theta$ respect to the x-axis (horizontal direction)

Image illustrating Components of a Vector Using the Angle It Makes Respect to x or y Axis

Components of this vector along x and y directions could be found as:

\begin{array}{l}v_x=|\vec{v}|\cos\theta\\v_y=|\vec{v}|\sin\theta\end{array}

We can find the length of this vector and the angle it makes respect to horizontal x-axis using the following equation:

Length of a vector in 2-D space is equal to:

\boxed{|\vec{v}|=\sqrt{v_x^2+v_y^2}}

The direction of a vector usually is reported by the angle it makes with positive $x$ -axis (Counter-clockwise) using following formula:

\boxed{\theta=\tan^{-1}\left(\frac{v_y}{v_x}\right)}

Example: Vector Connecting Two Points

If $A\left(1,2\right)$ and $B\left(-1,3\right)$ , calculate $\overrightarrow{AB}$ , length of $\overrightarrow{AB}$ and the angle it makes with positive $x$ -axis.

Solution:

$\begin{array}{rl}\overrightarrow{AB}=\overrightarrow{B}-\overrightarrow{A}\ = &\left(-1,3\right)-\left(1,2\right)\\=&(-1-1,3-2)=(-2,1)\end{array}$

Image illustrating Example: Vector Between Two Points

$\left|\overrightarrow{AB}\right|=\sqrt{\left(-2\right)^2+1^2}=\sqrt{5}$

$\ \theta=180-\tan^{-1}\left|\frac{AB_y}{AB_x}\right|=180-\tan^{-1}\left|\frac{1}{-2}\right|=153.43\degree$

A football is thrown with a horizontal velocity of 10m/s and a vertical velocity of 4.25 m/s. What is football's velocity?

9.68 m/s , 27 degrees

10.9 m/s , 23 degrees

12.45 m/s , 34 degrees

I don't know

Vectors and Their Components

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