Wize University Physics Textbook (Master) > Introduction to Vectors

Vectors and Their Components

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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 a⃗\vec{a}a in 2D space is shown as:
a⃗=(ax,ay)=axi^+ayj^\boxed{\vec{a}=\left( a_x,a_y\right )=a_x\hat{i}+a_y\hat{j} }



a=(ax​,ay​)=ax​i^+ay​j^​​
      

        - where vectors , i^\hat{i}i^ and j^\hat{j}j^​ are basic unit vectors in xxx and yyy directions, respectively. These unit vectors are:
i^=(1,0) j^=(0,1)\begin{array}{l}\hat{i}=(1,0)\\\ \hat{j}=(0,1)\end{array}i^=(1,0) j^​=(0,1)​
 Unit vectors are vectors with length equal to 1 which are used to show directions.
axa_xax​and aya_yay​are called components of vector a⃗\vec{a}a

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ٰ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:

Wize Tip
The length of the vector v⃗=[v1, v2]\vec{v}=\left[v_1,\ v_2\right]v=[v1​, v2​] is a scaler and is equal to:
∣v⃗∣=∣[v1, v2]∣=v1 2+v2 2|\vec v|=|[v_1,\ v_2]|=\sqrt{v_1^{\ 2}+v_2^{\ 2}}∣v∣=∣[v1​, v2​]∣=v1 2​+v2 2​​


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

Wize Tip
The length of the vector u⃗=[u1, u2, u3]\vec{u}=\left[u_1,\ u_2,\ u_3\right]u=[u1​, u2​, u3​] is
∣u⃗∣=∣[u1, u2, u3]∣=u1 2+u2 2+u3 2|\vec{u}|=|[u_1,\ u_2,\ u_3]|=\sqrt{u_1^{\ 2}+u_2^{\ 2}+u_3^{\ 2}}∣u∣=∣[u1​, u2​, u3​]∣=u1 2​+u2 2​+u3 2​​

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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 θ\thetaθrespect to the x-axis (horizontal direction)

Components of this vector along x and y directions could be found as:
vx=∣v⃗∣cos⁡θvy=∣v⃗∣sin⁡θ\begin{array}{l}v_x=|\vec{v}|\cos\theta\\v_y=|\vec{v}|\sin\theta\end{array}vx​=∣v∣cosθvy​=∣v∣sinθ​

Watch Out!
Do not always assume that "x is cosine, y is sine". This depends completely on where we define the angle θ\thetaθ. Normally, θ\thetaθis defined as an angle relative to the horizontal (x) axis, and these conventions will work, but when the angle is relative to the vertical (y) axis, the convention is flipped.

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:
∣v⃗∣=vx2+vy2\boxed{|\vec{v}|=\sqrt{v_x^2+v_y^2}}∣v∣=vx2​+vy2​​​
The direction of a vector usually is reported by the angle it makes with positive xxx-axis (Counter-clockwise) using following formula:
θ=tan⁡−1(vyvx)\boxed{\theta=\tan^{-1}\left(\frac{v_y}{v_x}\right)}θ=tan−1(vx​vy​​)​

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Example: Vector Connecting Two Points

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

Solution:

AB→=B→−A→ =(−1,3)−(1,2)=(−1−1,3−2)=(−2,1)\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}AB=B−A ==​(−1,3)−(1,2)(−1−1,3−2)=(−2,1)​

∣AB→∣=(−2)2+12=5\left|\overrightarrow{AB}\right|=\sqrt{\left(-2\right)^2+1^2}=\sqrt{5}​AB​=(−2)2+12​=5​

 θ=180−tan⁡−1∣AByABx∣=180−tan⁡−1∣1−2∣=153.43°\ \theta=180-\tan^{-1}\left|\frac{AB_y}{AB_x}\right|=180-\tan^{-1}\left|\frac{1}{-2}\right|=153.43\degree θ=180−tan−1​ABx​ABy​​​=180−tan−1​−21​​=153.43°

Exam Tip
*With these types of examples, instead of memorizing some general rule, it's best to draw the picture and reason about whether to add 180, subtract from 180, add 90, etc, depending on what you drew.

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

Practice: Magnitude of a Vector
An aircraft's location is 200 miles from base, at a direction of 30° W of S, at an  altitude of 35,000 ft. Find the distance between the aircraft and the base. 

Note: 1 mile = 5280 feet

Extra Practice

Practice: Magnitude of a Vector

Practice: Magnitude of a Vector
An aircraft's location is 200 miles from base, at a direction of 30° W of S, at an  altitude of 35,000 ft. Find the distance between the aircraft and the base. 
Note: 1 mile = 5280 feet

I'm moving and packed all my books into a box that weighs 17kg. I pull the box on a weightless dolly with a force of 60N at an angle of 30° to the horizontal surface. 
a) What is the acceleration of the box?  
b) The box feels the normal force from the ground and the force of gravity. Are the normal force and gravity an action-reaction pair? Explain your answer!

Vector arithmetic practice

Consider the vectors a⃗=−2i^+3j^−k^;   b⃗=4i^+3k^;   c⃗=−i^+3j^−k^\vec a=-2\hat i+3\hat j-\hat k; ~~~\vec b= 4\hat i+3\hat k; ~~~ \vec c=-\hat i+3\hat j-\hat ka=−2i^+3j^​−k^;   b=4i^+3k^;   c=−i^+3j^​−k^.
a) For each of these three vectors, find the magnitude.
b) For each of these three vectors, find a unit vector pointing in the same direction.

Component vectors practice

You try to pull a heavy box by attaching a rope and pulling with a force of 20.0 N at an angle of 13.4 degrees above horizontal. What are the components of this force in the horizontal and vertical directions? Draw a diagram to support your answer.

A rabbit, escaping a fox, runs 40o north of west at 10.0m/s. A coordinate system is established with the positive x-axis to the east and the positive y-axis to the north. Write the rabbit’s velocity in terms of components and unit vectors.

Break into components

Given a velocity of 5.0m/s at 40.0o above the -x axis:

Which of the following is an example of a vector quantity?

Which of the following operations will not change a vector?

Practice Vector Components

 Find horizontal and vertical components of a vector R shown in the Figure.  

A bowling ball with mass of 1 kg travelling at 0.75 m/s hits a 0.5 kg bowling pin, initially at rest. The bowling pin moves off with a speed of 0.25 m/s at an angle of 35º relative to the initial direction of the bowling ball. What is the final velocity of the bowling ball? (note: you cannot assume kinetic energy is conserved). 

Particle m1 with mass of 4×10−254 \times 10^{-25}4×10−25kg is travelling with a velocity of 5msi^5\frac{m}s\hat i5sm​i^ , and collides with particle m2 with mass 6×10−25 kg6\times 10^{-25}~\rm kg6×10−25 kg  which is initially at rest. After the interaction, m1 travels with speed of 5 m/s at an angle of 45 degrees with vector.
What is the total momentum of the system?
What is the vector velocity of m2?