Component form of a vector

Here you will learn about the component form of a vector, how to write a vector in component form including the meaning of the component form of a vector and how to use it to solve problems.

Students first learn about vectors in high school in a geometry class when they learn about translations and expand their knowledge as they progress through precalculus and calculus.

What is the component form of a vector?

The component form of a vector is the ordered pair of a vector that describes the movement or the change in the katex is not defined and katex is not defined values.

It is a way to represent a vector using ordered pairs, where the first value (katex is not defined-component) represents the horizontal movement or horizontal displacement and the second value (katex is not defined-component) represents the vertical movement or vertical displacement of the vector.

Vector katex is not defined which is pictured below.

Vector katex is not defined can be written in component form as katex is not defined

Notice how the katex is not defined-component is katex is not defined and the katex is not defined-component is katex is not defined

This is because from the initial point of the vector to the terminal point (or endpoint) of the vector, there is a horizontal movement of katex is not defined units right and a vertical movement that is katex is not defined units up.

Let’s look at how to calculate the component form of a vector.

Vector katex is not defined has an initial point of katex is not defined and a terminal point of katex is not defined Write vector katex is not defined in component form.

First, let’s look at the points on the coordinate plane.

Notice how from the initial point to the terminal point there is a horizontal movement of katex is not defined units right and then a vertical movement of katex is not defined units up.

The component form of vector katex is not defined is katex is not defined

Another way to calculate the component form of a vector is to use the formula.

Component form of a vector: katex is not defined

Notice how the katex is not defined-component represents the change in the katex is not defined-coordinates, and the katex is not defined-coordinate represents the change in the katex is not defined-coordinates.

That is exactly what you did when you determined the katex is not defined-component and katex is not defined-component on the graph.

Let’s use the formula to calculate the component form of vector katex is not defined

In this case, the initial point of vector katex is not defined is katex is not defined and the terminal point is katex is not defined

Initial point: katex is not defined

Terminal point: katex is not defined

Component form of vector katex is not defined

Notice how the notation of the component form has katex is not defined-shaped parentheses.

Parallel vectors

Vectors are considered to be parallel if they point in the same direction or exactly in opposite directions. Let’s look at two vectors that are in component form.

Vector katex is not defined

Vector katex is not defined

You know that lines are parallel if they have the same slope. It’s not much different for vectors. In component form, the katex is not defined-component is the change in katex is not defined and the katex is not defined-component is the change in katex is not defined

So, looking at vector katex is not defined the slope is katex is not defined and looking at vector katex is not defined the slope is katex is not defined

Now, let’s look at the vectors on the graph.

You can see that the vectors are parallel even though they are pointing in exactly opposite directions.

Vector katex is not defined and vector katex is not defined are also considered negative vectors because they are parallel and point in exactly opposite directions, katex is not defined

What is the component form of a vector?

Common Core State Standards

How does this relate to high school math?

• High School – The Real Number System – Vector and Matrix Quantities: HSN-VM.A.2 (+)
  Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point..

How to write a vector in component form from the graph

In order to write a vector in component form from a graph:

1. Count the units horizontally from the initial point to the terminal point.
2. Count the units vertically from the initial point to the terminal point.
3. Write the vector in component form.

Vector in component form examples

Example 1: vector on a graph

Write the component form of vector katex is not defined

1. Count the units horizontally from the initial point to the terminal point.

2Count the units vertically from the initial point to the terminal point.

3Write the vector in component form.

Since there are two units right, the katex is not defined-component is katex is not defined and since there are katex is not defined units up, the katex is not defined-component is katex is not defined

Vector katex is not defined

Example 2: vector on a graph

Write the component form of vector katex is not defined

Count the units horizontally from the initial point to the terminal point.

Show step

Count the units vertically from the initial point to the terminal point.

Show step

How to write a vector in component form using the formula

In order to write a vector in component form using the formula:

1. Identify katex is not defined and katex is not defined
2. Calculate the component form.
3. Write the vector in component form.

Example 3: calculate component form

Vector katex is not defined has an initial point katex is not defined and a terminal point katex is not defined Write the vector in component form.

Example 4: calculate component form

Vector katex is not defined has an initial point katex is not defined and a terminal point katex is not defined

Write the vector in component form.

How to find terminal or initial point of a vector

In order to find the terminal or initial point of a vector:

1. Identify the given information.
2. Use katex is not defined to find the missing point.
3. Write the point.

Example 5: find terminal point

Vector katex is not defined in component form is katex is not defined and has an initial point of katex is not defined

Find the terminal point.

Example 6: find initial point

Vector katex is not defined in component form is katex is not defined and has a terminal point katex is not defined

Find the initial point of vector katex is not defined.

Teaching tips for component form of a vector

• Have students discover component form by providing them with graphing activities where they can use the initial point and terminal point of a vector to determine its slope and then component form.

• Incorporate investigative activities on digital graphing platforms.

• Have struggling students use digital platforms that have embedded tutorial videos.

• Instead of worksheets, have students game play to practice skills and concepts.

Easy mistakes to make

• Mixing up the katex is not defined-component and katex is not defined-component
  When writing a vector in component form, like ordered pairs, write the horizontal movement first (katex is not defined-component) and vertical movement second (katex is not defined-component).

• Confusing the terminal point with the initial point
  For example, when writing the vector katex is not defined with initial point katex is not defined and terminal point katex is not defined in component form, calculating it incorrectly.
  
  katex is not defined instead of subtracting the initial point from the terminal point. The correct way to calculate is katex is not defined
  
  Unlike finding slope, the order of katex is not defined is essential to the representation of the vector.
  
  From the graph below, you can see how the vectors are parallel but they are not the same. Further confirming that the order of subtraction is essential to the correct component form of a vector.
  
  

• Thinking that vectors pointing in different directions are not parallel
  Vectors are parallel as long as their slopes are the same. They do not have to be pointing in the same direction to be parallel.

• Forgetting to use the katex is not defined shaped brackets
  The component form of a vector is not an ordered pair, it represents the horizontal and vertical movement so using the katex is not defined shaped brackets indicates it is not an ordered pair.

Related vector lessons

• Subtracting vectors
• Adding vectors
• Vector multiplication
• Magnitude of a vector
• Vector practice problems

The next lessons are

• Lines
• Angles
• Angles in parallel lines
• Trigonometric identities
• Pythagorean Theorem
• Magnitude of a vector
• Direction of a vector
• Vector addition

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