Fundamentals of Vectors
2. Vector Components
Learning outcomes
- I can resolve vectors into horizontal and vertical components.
- I can determine vector components using trigonometry.
- I can reconstruct vectors from their components.
- I can interpret component notation.
- I can solve practical problems involving vector components.
What Are Vector Components?
A vector has both magnitude and direction.
Sometimes it is easier to work with a vector by splitting it into two simpler vectors called components.
For a two-dimensional vector, we normally use:
- A horizontal component along the x-axis.
- A vertical component along the y-axis.
Together, these two components have the same overall effect as the original vector.
For example, a force acting diagonally upward can be separated into:
- A force acting horizontally.
- A force acting vertically.
This process is called resolving a vector into components.
Understanding the Vector Triangle
When a vector is resolved into horizontal and vertical components, the three vectors form a right-angled triangle.
Suppose a vector V
has magnitude V and makes an angle θ above the horizontal.
We can label:
- V = magnitude of the original vector
- Vx = horizontal component
- Vy = vertical component
- θ = angle measured from the horizontal
The original vector is the hypotenuse of the triangle.
The horizontal and vertical components are the other two sides.
This allows us to use trigonometry to calculate the components.
Resolving a Vector Using Trigonometry
Remember the trigonometric relationships:
If the angle is measured from the horizontal, the horizontal component is adjacent to the angle and the vertical component is opposite.
Therefore:
Vx = Vcosθ
and
Vy = Vsinθ
These are two of the most important equations when working with vectors.
x = rcosθ, y = rsinθ
Remember
If the angle is measured from the horizontal:
Horizontal → cosine
Vertical → sine
Worked Example 1: Finding Components
A force of 50 N acts at an angle of 30∘ above the horizontal.
Find the horizontal and vertical components.
Step 1: Identify the information
Step 2: Find the horizontal component
Step 3: Find the vertical component
Therefore, the vector has components:
Vx = 43.3 N Vy = 25.0 NComponent Notation
Vectors can also be written using component notation.
For example:
This means:
The first number represents the horizontal component, and the second represents the vertical component.
Another common notation is:
where:
- i represents the positive x-direction.
- j represents the positive y-direction.
Both forms describe the same vector.
Positive and Negative Components
The signs of the components tell us the direction.
For the horizontal component:
- Positive x → right
- Negative x → left
For the vertical component:
- Positive y → up
- Negative y → down
For example:
means the vector points:
- 4 units to the left
- 7 units up
Therefore, the vector points generally up and to the left.
Reconstructing a Vector
Sometimes we know the horizontal and vertical components and need to determine the original vector.
Suppose:
and
Because the components form a right-angled triangle, we can use the Pythagorean theorem:
Therefore:
Substituting the values:
The magnitude of the original vector is 10 m/s.
Finding the Direction
We can also use the components to determine the vector's direction.
Using:
we can calculate:
For our vector:
Therefore:
The complete vector is therefore:
10 m/s at 53.1o above the horizontalResolving Forces
Vector components are particularly important when working with forces.
Imagine a person pulling a box using a rope at an angle.
Suppose the person pulls with:
30o above the horizontal.The horizontal component is:
The vertical component is:
The horizontal component helps move the box forward, while the vertical component pulls slightly upward on the box.
Vector Components in Projectile Motion
Vector components are also important when studying objects moving through the air.
Imagine a football kicked at an angle.
Its initial velocity can be separated into:
- Horizontal velocity
- Vertical velocity
Suppose a ball is launched at: 20 m/s at an angle of: 40o
The horizontal velocity is:
The vertical velocity is:
These components can then be analysed separately when studying the ball's motion.
Components in Navigation
Vector components are also useful in navigation.
Suppose a boat travels northeast.
Its velocity can be separated into:
- An eastward component.
- A northward component.
Similarly, wind acting on an aircraft can be separated into horizontal directions so that pilots can calculate the aircraft's actual motion.
A Useful Problem-Solving Method
When solving vector-component problems:
1. Draw the vector
Sketch the vector and its direction.
2. Draw the components
Create horizontal and vertical arrows to form a right triangle.
3. Identify the angle
Check carefully whether the angle is measured from the horizontal or vertical.
4. Choose the correct trigonometric relationship
If the angle is measured from the horizontal:
5. Calculate
Substitute the values and include the correct units.
6. Check the signs
Make sure the signs of the components match their directions.
Be Careful: Where Is the Angle?
You should not automatically assume that cosine always gives the horizontal component.
The formulas
work when θ is measured from the horizontal.If the angle is measured from the vertical, the relationships switch.
For example:
The safest approach is to identify:
- Hypotenuse → original vector
- Adjacent → use cosine
- Opposite → use sine
Then use SOH CAH TOA rather than simply memorizing which component uses sine or cosine.
Worked Example 2: Practical Problem
A rescue helicopter travels at 80 m/s at an angle of 25∘ north of east.
Horizontal component
Vertical component
Therefore, the helicopter's velocity can be written as:
\( \vec{v} \) = ⟨72.5, 33.8⟩ m/sThis tells us that the helicopter is travelling approximately:
- 72.5 m/s east
- 33.8 m/s north
Did You Know?
Engineers and computer programmers often work with vectors by using their components rather than their magnitude and direction.
Computer games, flight simulators, GPS systems, robotics, and 3D animation all use vector components to calculate how objects move through space.
Key Terms
Vector component – One part of a vector acting in a particular direction.
Horizontal component – The part of a vector acting along the x-axis.
Vertical component – The part of a vector acting along the y-axis.
Resolve – To separate a vector into its components.
Reconstruct – To combine components to determine the original vector.
Magnitude – The size of a vector.
Component notation – A way of describing a vector using its horizontal and vertical components, such as ⟨6,8⟩.
Key Takeaways
- A vector can be resolved into horizontal and vertical components.
- The components form a right-angled triangle with the original vector.
- If the angle is measured from the horizontal: Vx = Vcosθ Vy = Vsinθ
- A vector can be reconstructed using: \( V = \sqrt[]{V_x^2 + V_y^2} \)
- Its direction can be found using: \( \theta = tan^{-1}( \frac{V_y}{V_x}) \)
- Positive and negative component values indicate different directions.
- Always check where the angle is measured from before choosing sine or cosine.
- Vector components are used extensively in forces, projectile motion, navigation, engineering, and computer simulations.