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Addition of Vectors

Lesson

Vectors can be added together.  To do this, we use what is called the top to tail method.  We put the head of the second vector on top of the tail of the first and the resultant vector (of the result of adding the two vectors together) is the vector that reaches from the initial point of the first, to the terminal point of the second. So adding vectors u and v below, results in w

So geometrically this is what is happening with vector addition, let's now look at the algebraic process.  

For this I am going to use the column vector notation.  Remember that the first element in the column matrix indicates the x movement (right is positive and left is negative) and the second element indicates the change in y, (up positive and down negative).  

Using column vector notation, makes adding vectors a relatively simple process akin to adding column matrices.  We just added the corresponding elements together.  

Let's look at what happened geometrically with the same vector addition.

Here vectors $u$u and $v$v are drawn.

 

We can see that the matrix components above, relate directly to the $x$x and $y$y components of these vectors. 

To add together, we reposition vector $v$v so that the head of $v$v, is positioned at the tail of $u$u

The resultant vector is the vector that extends from the beginning of $u$u, and ends at the tail of $v$v

This vector has $x$x and $y$y components as shown here. 

And this is the same as the matrix result we had earlier. 

 

Using column vector notation, makes adding vectors a relatively simple process akin to adding column matrices.  We just added the corresponding elements together.  Thus, using this process, we can add any number of vectors together. 

This applet will allow you to create vectors of given sizes, and then demonstrate their additions geometrically.  

 

Worked Examples

Question 1

Which vector is the result of $a+b$a+b?

Loading Graph...

  1. $c$c

    A

    $d$d

    B

    $c$c

    A

    $d$d

    B

Question 2

Consider the two vectors $a$a and $b$b, as graphed below.

  1. Plot the result of $a+b$a+b on the same axes.

    Loading Graph...

  2. Now plot the result of $b+a$b+a.

    Loading Graph...

  3. Are the two resulting vectors $a+b$a+b and $b+a$b+a equal?

    Yes

    A

    No

    B

    Yes

    A

    No

    B

Question 3

Consider the plotted vectors.

Loading Graph...

  1. State the vectors using column vector notation.

    $a$a $=$=
        $\editable{}$    
        $\editable{}$    
    $b$b $=$=
        $\editable{}$    
        $\editable{}$    
    $c$c $=$=
        $\editable{}$    
        $\editable{}$    
  2. Find $a+b$a+b.

    $a+b$a+b $=$=
        $\editable{}$    
        $\editable{}$    
  3. Find $c+b+a$c+b+a.

    $c+b+a$c+b+a $=$=
        $\editable{}$    
        $\editable{}$    

 

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