Unit 1.1 | Understanding vectors and the standard units used in Physics

Why Vectors

You might be wondering where and why we use vectors in physics.

Using vectors allows us to view motion in multiple directions.

Look at the image below. The soccer ball is kicked at a velocity [katex]v_0[/katex] that is 30º above the soccer field.

Note that velocity means speed with a direction, and [katex]v_0[/katex] is the variable for “initial velocity.” It is a placeholder for the numerical value of the velocity of the soccer ball.

\(v_0\)

\(30^\circ\)

\(H\)

\(R\)

Physicists can take the [katex]v_0[/katex] velocity vector and separate it into two parts: the [katex]H[/katex] vector and the [katex]R[/katex] vector as shown in the image above. These are what we call vector components.  

Vector Components

Since we can break 1 vector down into 2 components of this motion, we can see that the ball is really moving both up and to the right at the same time.

We can use these components (or parts of the velocity vector) to solve real world problems, like how far did the ball travel before landing on the ground?

LRN Vector Math

In Physics, there are 4 things you will be doing with vectors. The most important one is the last one and there is a short video below showing how to do it.

1. [katex] number \times \vec {A} [/katex] : Multiply a vector by a scalar to increase the size of the vector.
2. [katex] \vec{A} + \vec{B} [/katex] : Add two vectors along the same axis to get a resultant vector.
3. [katex] \vec{A} + \vec{B} [/katex] : Add two perpendicular vectors to get a resultant vector.
4. [katex] \vec{A} = \vec{A_x} + \vec{A_y}[/katex] : Take 1 vector and break it down into the horizontal and vertical components

\(\vec{A}\)

\(2\vec{A}\)

\(\vec{A}\)

\(\vec{B}\)

\(\vec{A}+\vec{B}\)

\(\vec{A}\)

\(\vec{B}\)

\(R^2=A^2+B^2\)

\(\theta\)

\(\tan\theta=\tfrac{B}{A}\)

\(\vec{A}\)

\(A_x = A\cos\theta\)

\(A_y = A\sin\theta\)

\(\theta\)

Okay now that you know what we can do with vectors, let’s put it into practice! Watch the short video below, and try the example problems after.

PQ Vector Components

After watching the video above you should have a solid understanding of adding vectors and breaking them into components. Attempt all the practice questions below and be sure to understand all missed ones before moving on.

(1) Find the resultant vector (magnitude AND direction) given the following components:

1. A_x = 5.7, A_y = 3.2; Answer: (A = 6.5 at 29.3 degrees above the +x axis)
2. B_x = -10, B_y = -3; Answer: (B = 10.4 at 16.7 degrees below the -x axis)
3. C_x = 11.65, C_y = -20.17; Answer: (C = 23.3 at 60 degrees below the +x axis)

(2) Find the X and Y components of the following vectors:

1. 35 m/s at 57° from the x-axis; Answer: (X: 19.1 m/s, Y: 29.4 m/s)
2. 12 m/s at 34° S of W; Answer: (X: -10 m/s, Y: -6.7 m/s)
3. 20 m/s 275° from the x-axis; Answer: (X: 1.75 m/s, Y: -20 m/s)
4. A plane takes off with a velocity, v, of 40 m/s at an angle of 15° above the ground; Answer: (v_x = 38.6 m/s, v_y: 10.4 m/s)

(3) Challenge: Listed below are 4 distance vectors.

• A: 15 m at 60° above the +x-axis
• B: 10 m east
• C: 25 m south
• D: 50 meters at 28° North of West

Using the vectors above, perform the following operations:

1. A + B + C + D; Answer: (29 m at 23° above the -x-axis or 23° N of W)
2. A – C; Answer: (38.7 m at 78.8° above the +x-axis or 78° N of E)
3. D + 2C; Answer: (51.4 m at 31° below the -x-axis or 31° S of W)

Hint: Draw out each vector on a graph. Break each vector into x and y components, then perform the operations in each direction.

Using the Standard Units of Measurement

If you’re from the United States you may be used to seeing miles, miles per hour, feet, pounds etc.

This is the imperial system of measurement. And it makes calculations and conversions difficult.

Instead, in physics, we use the metric system (kilometers, meters, kilograms, etc) that the rest of the world is familiar with.

Why Metric?

The metric system is much easier to convert between orders of magnitude.

For example, how many centimeters are in a meter? Well we can simply look at the prefix “centi” and tell that there are 100 centimeters in 1 meter. 

So in Physics, you’ll commonly see the following standard units: 

• Distance is measured in meters (m)
• Speed is measured in meters per second (m/s)
• Mass (not weight) is measured in kilograms (kg)
• Time is measured in seconds (s)

From now on, when solving a problem, make sure to convert all given numbers to the standard units above!

Unit Conversions

As mentioned earlier, converting units is pretty easy in the metric system.

You just have to memorize what each prefix means. The chart below will help with that.

As you do more Physics problems, unit conversions will become super easy!

AND if you are having trouble converting units, watch this simple video explanation. 

\(1000\)

\(100\)

\(10\)

\(1\)

\(0.1\)

\(0.01\)

\(0.001\)

PQ Converting Units

Solve ALL problems before checking your answer below.

1. Convert 320 grams to kg
2. Convert 145 mm to meters
3. Convert 100 km/hr the standard unit of velocity used in Physics
4. Convert 50 cm/min to m/s
5. Convert 365 days to seconds
6. Given that 1 inch = 2.54 cm convert 4674 inches to meters
7. The volume of a steel cube is 300 cm³. Convert this to m³
8. The density of a iron sphere is 7.9 g/cm³. Convert this to kg/m³

Click Here For All Answers:

1. .32 kg
2. .145 m
3. 27.8 m/s (In Physics we use ‘ m/s ‘ as a unit of speed and velocity)
4. .0083 m/s
5. 3.15 x 10⁷ s or 31500000 s
6. 118.7 m
7. 3 x 10^-4 m³ or .0003 m³
8. 7900 kg/m³

Lesson 1.2 Preview

In the next lesson we will begin to cover kinematics by exploring the 3 kinematic variables: displacement, velocity, and acceleration. We will use standard units. In lesson 1.4 we will begin to utilize our knowledge of vectors to real world problems. ## Attribution Guidelines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarizing, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-1-1-understanding-vectors-and-the-standard-units-used-in-physics/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-1-1-understanding-vectors-and-the-standard-units-used-in-physics/