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 x and y values.

It is a way to represent a vector using ordered pairs, where the first value ( x -component) represents the horizontal movement or horizontal displacement and the second value ( y -component) represents the vertical movement or vertical displacement of the vector.

Vector \vec{v} which is pictured below.

Vector \vec{v} can be written in component form as \vec{v}=\langle{8, \, 4}\rangle.

Notice how the x -component is 8 and the y -component is 4.

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 8 units right and a vertical movement that is 4 units up.

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

Vector \vec{AB} has an initial point of (- \, 4, \, 3) and a terminal point of (- \, 1, \, 7). Write vector \vec{AB} 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 3 units right and then a vertical movement of 4 units up.

The component form of vector \vec{AB} is \langle{3, \, 4}\rangle.

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

Component form of a vector: \langle{x_2-x_1, \, y_2-y_1}\rangle

Notice how the x -component represents the change in the x -coordinates, and the y -coordinate represents the change in the y -coordinates.

That is exactly what you did when you determined the x -component and y -component on the graph.

Let’s use the formula to calculate the component form of vector \vec{AB}.

In this case, the initial point of vector \vec{AB} is A \, (- \, 4, \, 3) and the terminal point is B \, (- \, 1, \, 7).

Initial point: \left(x_1, \, y_1\right)=A \, (- \, 4, \, 3)

Terminal point: \left(x_2, \, y_2\right)=B \, (- \, 1, \, 7)

Component form of vector \vec{AB}=\langle{- \, 1-(- \, 4), \, 7-3}\rangle \rightarrow \langle{3, \, 4}\rangle

Notice how the notation of the component form has v -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 \vec{v}=\langle{- \, 5, \, 1}\rangle

Vector \vec{w}=\langle{5, \, - \, 1}\rangle

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

So, looking at vector \vec{v}, the slope is \cfrac{1}{- \, 5} and looking at vector \vec{w}, the slope is \cfrac{- \, 1}{5}.

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 \vec{v} and vector \vec{w} are also considered negative vectors because they are parallel and point in exactly opposite directions, \vec{v}=- \, \vec{w}.

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 \vec{AB}.

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 x -component is 2, and since there are 6 units up, the y -component is 6.

Vector \vec{AB}=\langle{2, \, 6}\rangle.

Example 2: vector on a graph

Write the component form of vector \vec{CD}.

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

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

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 \bf{\left(\textbf{x}_1, \, \textbf{y}_1\right)} and \bf{\left(\textbf{x}_2, \, \textbf{y}_2\right)}.
2. Calculate the component form.
3. Write the vector in component form.

Example 3: calculate component form

Vector \vec{AB} has an initial point A \, (- \, 3, \, 7) and a terminal point B \, (- \, 1, \, - \, 1). Write the vector in component form.

Example 4: calculate component form

Vector \vec{v} has an initial point (0, \, 6) and a terminal point (10, \, - \, 1).

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 \bf{\langle{\textbf{x}_2-\textbf{x}_1, \, \textbf{y}_2-\textbf{y}_1}\rangle} to find the missing point.
3. Write the point.

Example 5: find terminal point

Vector \vec{w} in component form is \langle{- \, 9, \, 5}\rangle and has an initial point of (9, \, - \, 4).

Find the terminal point.

Example 6: find initial point

Vector \vec{u} in component form is \langle{- \, 6, \, - \, 3}\rangle and has a terminal point (- \, 10, \, - \, 10).

Find the initial point of vector \vec{u} .

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 \textbf{x} -component and \textbf{y} -component
  When writing a vector in component form, like ordered pairs, write the horizontal movement first ( x -component) and vertical movement second ( y -component).

• Confusing the terminal point with the initial point
  For example, when writing the vector \vec{AB} with initial point (8, \, 2) and terminal point (- \, 5, \, 3) in component form, calculating it incorrectly.
  
  \langle{8-(- \, 5), \, 2-3}\rangle=\langle{13, \, - \, 1}\rangle, instead of subtracting the initial point from the terminal point. The correct way to calculate is \langle{- \, 5-8, \, 3-2}\rangle=\langle{- \, 13, \, 1}\rangle.
  
  Unlike finding slope, the order of \langle{x_{2}-x_{1}, \, y_{2}-y_{1}}\rangle 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 \textbf{"v"} shaped brackets
  The component form of a vector is not an ordered pair, it represents the horizontal and vertical movement so using the “v” 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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