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Addition and subtractionCoordinate grid

Integers Magnitude of a vectorHere you will learn about adding vectors, including what it is and how to do it.

Students will first learn about adding vectors as part of the number system in high school.

**Adding vectors** is adding one vector to another vector. This is sometimes known as a vector sum.

As vectors can be located anywhere in a space, the start of the vector is called the **tail**, and the end of the vector is called the **head**. The direction of the vector is therefore from the tail to the head and is indicated using an **arrow** through the middle of the vector.

Here, the arrow of vector \vec{v}=\langle{3,2}\rangle points from the tail to the head of the vector.

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DOWNLOAD FREELet’s look at some examples of vectors.

For the addition of vectors, \vec{a}+\vec{b}, place the tail of vector \vec{b} onto the head of vector \vec{a}.

Adding the vectors forms the new vector \vec{a}+\vec{b}. This is drawn as a straight line from the tail of \vec{a} to the head of \vec{b}.

Now you can identify the components of the new vector. Notice they form the base and height of a right triangle, in which the vector is the hypotenuse.

Here, the horizontal x component of the vector \vec{a}+\vec{b} is 4 right and the vertical y component is 6 up. So \vec{a}+\vec{b}=\langle{4,6}\rangle.

Though it can be helpful, when finding a vector sum you do not have to draw a diagram. You can just add the vector components. First, add the x components (first numbers) and then add the y components (second numbers).

\begin{aligned}\vec{a}+\vec{b}&=\langle{3,2}\rangle+\langle{1,4\rangle} \\\\ &=\langle{3+1,2+4}\rangle \\\\ &=\langle{4,6}\rangle \end{aligned}The new vector that results from adding two given vectors is known as the **resultant vector**.

The addition of vectors is **commutative** which means that the order in which you add the vectors is not important. In other words we will get the same result if you add the second vector to the first vector.

The diagram is a **parallelogram**. The resultant vector is the diagonal of the parallelogram.

Note: This page covers only 2 -dimensional vectors.

How does this relate to high school math?

**The Number System (HSN.VM.B.4a)**Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

**The Number System (HSN.VM.B.4b)**

Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

In order to add vectors:

**Add the \textbf{x} components.****Add the \textbf{y} components.****Write the resultant vector.**

Add the given vectors, \langle{2,4}\rangle+\langle{5,1}\rangle, shown on the coordinate plate.

Notice the direction of the vectors is positive.

**Add the \textbf{x} components.**

Connect the tail of vector \vec{a} to the head of vector \vec{b} to show the resultant vector \vec{a}+\vec{b}.

Look at the horizontal change from the tail of the first vector to the head of the second vector.

The horizontal x component is 7 right, or 7.

2**Add the \textbf{y} components.**

Look at the vertical change from the tail of the first vector to the head of the second vector.

The vertical y component is 5 up, or 5.

3**Write the resultant vector.**

Write the resultant vector \vec{a}+\vec{b} in the form \langle{x,y}\rangle.

**Add the \textbf{x} components.**

Draw a straight line connecting the tail of \vec{a} to the head of \vec{b} and draw an arrow to show the direction of the vector \vec{a}+\vec{b}.

Look at the horizontal change from the tail of the first vector to the head of the second vector.

**Add the \textbf{y} components.**

Look at the vertical change from the tail of the first vector to the head of the second vector.

**Write the resultant vector.**

\langle{2,5}\rangle

Add the given vectors: \langle{8,2}\rangle+\langle{7,1}\rangle+\langle{3,0}\rangle

**Add the \textbf{x} components.**

Add the first numbers of the original vectors:

8+7+3=18

**Add the \textbf{y} components.**

Add the second numbers of the original vectors:

2+1+0=3

**Write the resultant vector.**

Write the two answers in the component form of a vector:

\langle{8,2}\rangle+\langle{7,1}\rangle+\langle{3,0}\rangle=\langle{18,3}\rangle

Add the given vectors: \langle{- \, 5,- \, 2}\rangle+\langle{2,- \, 4}\rangle

**Add the \textbf{x} components.**

Add the first numbers of the original vectors:

- \, 5+2=- \, 3

**Add the \textbf{y} components.**

Add the second numbers of the original vectors:

- \, 2+(- \, 4)=- \, 2-4=- \,6

**Write the resultant vector.**

Write the two answers in the component form of a vector:

\langle{- \, 5,- \, 2}\rangle+\langle{2,- \, 4}\rangle=\langle{- \, 3,- \, 6}\rangle

Let \vec{a}=\langle{7,3}\rangle and \vec{b}=\langle{- \, 3,- \, 4}\rangle. Calculate 2\vec{a}+\vec{b} writing your answer in the component form of the vector.

**Add the \textbf{x} components.**

The vector 2\vec{a} is the same as \vec{a}+\vec{a} so 2\vec{a}+\vec{b} is the same as \vec{a}+\vec{a}+\vec{b}.

Add the first numbers of the original vectors:

7+7+(- \, 3)=14-3=11

**Add the \textbf{y} components.**

Add the second numbers of the original vectors:

3+(- \, 4)+(- \, 4)=3-4-4=- \, 5

**Write the resultant vector.**

Write the two answers in the component form of the vector:

2\vec{a}+\vec{b}=\langle{11,- \, 5}\rangle

Let \vec{a}=\langle{- \, 3,- \, 1}\rangle,~\vec{b}=\langle{- \, 2,4}\rangle and \vec{c}=\langle{5,- \, 3}\rangle. Find \vec{a}+\vec{b}+\vec{c}. Write your answer in the component form.

**Add the \textbf{x} components.**

Add the first numbers of the original vectors:

- \, 3+(- \, 2)+5=0

**Add the \textbf{y} components.**

Add the second numbers of the original vectors:

- \, 1+4-3=0

**Write the resultant vector.**

Write the two answers in the component form of the vector:

\langle{- \, 3,- \, 1}\rangle+\langle{- \, 2,4}\rangle+\langle{5,- \, 3}\rangle=\langle{0,0}\rangle

- Instead of jumping to the rule, let students explore adding vectors using the head-to-tail method on coordinate system graphs. This will help them understand why the rule for adding vectors work and in turn help them have a deeper understanding of vectors.

- Make sure that students understand the vocabulary for the components of a vector. Terms such as tail (starting point), head (end point), magnitude and direction should be clearly defined for students.

**Incorrectly adding negative integers**

Mistakes can easily be made when adding a positive number and a negative number. Make sure that students’ negative numbers skills are well practiced before teaching vectors. For example, 3+(- \, 4)=3-4=- \, 1

**Corresponding components with the wrong axes**

The horizontal components are related to the x -axis and the vertical components y -axis. Show students how this connects to their knowledge of the coordinate grid.

**Confusing scalars, vectors and matrices**

Since this is an introductory topic, students may easily confuse terms. Clearly give examples of each and correct students when they use these terms incorrectly.

- Component form of a vector
- Subtracting vectors
- Vector multiplication
- Vector practice problems

1. Add the given vectors, \vec{a}=\langle{3,3}\rangle and \vec{b}=\langle{5,3}\rangle, shown on the coordinate plane.

\langle{6,8}\rangle

\langle{- \, 2,0}\rangle

\langle{8,6}\rangle

\langle{3,3}\rangle

Adding the vectors forms the new vector \vec{a}+\vec{b}.

Calculate the horizontal and vertical component of the vector by counting squares:

The resultant vector is \langle{8,6}\rangle.

2. Find the resultant vector of \langle{5,1}\rangle+\langle{3,6}\rangle, shown on the coordinate plane. Write your answer in the component form of the vector.

\langle{2,4}\rangle

\langle{- \, 3,0}\rangle

\langle{7,8}\rangle

\langle{0,- \, 3}\rangle

Connect the tail of \vec{a} to the head of \vec{b}.

Look at the horizontal and vertical change in vector \vec{a}+\vec{b}.

The horizontal component is – \, 3.

The vertical component is 0.

The resultant vector is \langle{- \, 3,0}\rangle.

3. Add the given vectors, \langle{0,4}\rangle+\langle{- \, 4,0}\rangle, shown on the coordinate plane.

\langle{4,4}\rangle

\langle{- \, 4,- \, 4}\rangle

\langle{- \, 4,4}\rangle

\langle{4,- \, 4}\rangle

Adding the vectors forms the new vector \vec{a}+\vec{b}.

Look at the horizontal and vertical change from the tail of the first vector to the head of the second vector.

The horizontal component is 4.

The vertical component is – \, 4.

The resultant vector is \langle{4,- \, 4}\rangle.

4. Add vector \vec{r} and vector \vec{s}.

\vec{r}=\langle{3,2}\rangle

\vec{s}=\langle{1,5}\rangle

Write the resultant vector in the component form.

\langle{3,10}\rangle

\langle{2,- \, 3}\rangle

\langle{4,7}\rangle

\langle{4,- \, 3}\rangle

\langle{3,2}\rangle+\langle{1,5}\rangle=\langle{3+1,2+5}\rangle=\langle{4,7}\rangle

5. Calculate the resultant vector

\langle{4,5}\rangle+\langle{1,- \, 3}\rangle+\langle{- \, 2,1}\rangle

\langle{7,9}\rangle

\langle{10,- \, 4}\rangle

\langle{5,2}\rangle

\langle{3,3}\rangle

\begin{aligned}&\langle{4,5}\rangle+\langle{1,- \,3}\rangle+\langle{- \, 2,1}\rangle \\\\
&=\langle{4+1+(- \, 2),5+(- \, 3)+1}\rangle \\\\
&=\langle{3,3}\rangle \end{aligned}

6. Let \vec{a}=\langle{1,4}\rangle and \vec{b}=\langle{2,- \, 3}\rangle. Calculate \vec{a}+2\vec{b}.

\langle{- \, 3,- \, 2}\rangle

\langle{5,- \, 2}\rangle

\langle{4,- \, 6}\rangle

\langle{4,5}\rangle

\begin{aligned}&\langle{1,4}\rangle+\langle{2,- \, 3}\rangle+\langle{2,- \, 3}\rangle \\\\
&=\langle{1+2+2,4+(- \, 3)+(-3)}\rangle \\\\
&=\langle{5,- \, 2}\rangle \end{aligned}

Subtracting vectors is similar, but instead of adding the components, you subtract them.

Derived from the Pythagorean theorem, you square the components of the vector and then take their square root.

Yes, though the examples on this page only show the sum of two vectors, any number of vectors can be added.

Vectors that have a magnitude of 1.

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