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Math help from the Learning Centre

This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

Vector Addition

The simplest vector operation is vector addition, It's done by placing the tail of one vector at the tip of the other, I.e., add the vectors "tip to tail". We then draw a new vector from the start (tail) of your first vector to the end (tip) of your last vector.

Adding vectors is useful for when you want to know what the overall effect of multiple vectors will be. For example, a moving boat travelling on moving waters, a collision between 2 objects, a plane flying through strong winds, etc.

 

Example 1: A sailboat is traveling on calm (non-moving) waters at a speed of \(  \vec{v}_{\,boat} = 40km/h \) \( east \) and the wind is pushing it at a speed of \(  \vec{v}_{\,wind} = 10km/h \) \( north \). What is the combined vector that the boat is travelling on?

Solution:

Because the boat is travelling east and the wind is pushing the boat north at the same time, these effects will combine and cause the boat to travel east and north.

1. We can see this graphically if we add the vectors "tip to tail". In this case, I added \( \vec{v}_{\,wind} \) to the tip of \( \vec{v}_{\,boat} \). We then draw our new vector, \( \vec{v}_{\,result} \), from the tail of \( \vec{v}_{\,boat} \) to the tip of \( \vec{v}_{\,wind} \).

Mathematically, this looks like: \( \vec{v}_{\,boat} \) \( + \) \( \vec{v}_{\,wind} \) \( = \) \( \vec{v}_{\,result} \)

In the graph below, we have north as the +y axis and east as the +x axis so the directions match those of a compass.

2. You can also see that if we add \( \vec{v}_{\,boat} \) to the tip of \( \vec{v}_{\,wind} \), we will get the same \( \vec{v}_{\,result} \) as before, so the order of addition doesn't matter.

3. Now we need to find \( \vec{v}_{\,result} \) in rectangular coordinates. To do this, we need to write the x and y components of \( \vec{v}_{\,result} \). Because \( \vec{v}_{\,boat} \) is only effecting the x axis (east) speed, the x component of \( \vec{v}_{\,result} \), which we will label \( v_{\,result} \) \( _{\,x} \), will be equal to \( \vec{v}_{\,boat} = 40km/h \).

Similarly, \( \vec{v}_{\,wind} \) is only effecting the y axis (north) speed, so the y component of \( \vec{v}_{\,result} \), labeled \( v_{\,result} \) \( _{\,y} \), will be equal to \( \vec{v}_{\,wind} = 10km/h \).

Finally, we just need to write \(  \vec{v}_{\,result} \) in point form like this: \(  \vec{v}_{\,result} \) \( = \) (\( \vec{v}_{\,boat} \),\( \vec{v}_{\,wind} \)) \( = \) (\( 40km/h \),\( 10km/h \))

 

Example 2: A plane traveling along the vector \(  \vec{v}_{\,plane} = (145km/h,-91km/h) \) is flying through a strong wind which is blowing along the vector \(  \vec{v}_{\,wind} = (-32km/h,-18km/h) \). Add these vectors to find what vector the plane is actually travelling along, \(  \vec{v}_{\,result} \), in rectangular coordinates.

Solution:

This is a slightly trickier example because both vectors have x and y components.

1. To start solving this, we can plot our \( \vec{v}_{\,plane} \) and \( \vec{v}_{\,wind} \) vectors and then move one to the tip of the other, like this:

2. Next, we can draw our new vector, \(  \vec{v}_{\,result} \), from the start of \( \vec{v}_{\,plane} \) to the end of \( \vec{v}_{\,wind} \) like so:

3. Now, we need to find the x and y components of \( \vec{v}_{\,result} \). To do this, it is easier to graph \( \vec{v}_{\,plane} \) and \( \vec{v}_{\,wind} \) as their x and y components, then add the x components and the y components to get \( v_{\,result} \) \( _{\,x} \) and \( v_{\,result} \) \( _{\,y} \). You can see this graphically in the following plot:

4. Now we just have to add the components. Starting with the x components, we have \( v_{\,plane} \) \( _{\,x} \) and \( v_{\,wind} \) \( _{\,x} \) that we can add together to get \( v_{\,result} \) \( _{\,x} \) like this:

\( v_{\,result} \) \( _{\,x} \) \( = \) \( v_{\,plane} \) \( _{\,x} \) \( + \) \( v_{\,wind} \) \( _{\,x} \)

\( v_{\,result} \) \( _{\,x} \) \( = 145 + (-32) \)         

\( v_{\,result} \) \( _{\,x} \) \( = 113 \)                         

 

5. Similarly, we can get \( v_{\,result} \) \( _{\,y} \) from \( v_{\,plane} \) \( _{\,y} \) and \( v_{\,wind} \) \( _{\,y} \) like this:

\( v_{\,result} \) \( _{\,y} \) \( = \) \( v_{\,plane} \) \( _{\,y} \) \( + \) \( v_{\,wind} \) \( _{\,y} \)

\( v_{\,result} \) \( _{\,y} \) \( = (-91) + (-18) \)    

\( v_{\,result} \) \( _{\,y} \) \( = -109 \)                     

 

6. Now, we can combine our \( v_{\,result} \) \( _{\,y} \) and \( v_{\,result} \) \( _{\,y} \) components to get \( \vec{v}_{\,result} \) like this:

\( \vec{v}_{\,result} \) \( = \) (\( v_{\,result} \) \( _{\,x} \),\( v_{\,result} \) \( _{\,y} \))

\( \vec{v}_{\,result} \) \( = (113,-109) \)         

Addition Summary

In summary, to add vectors you need to:

  • graph the vectors and move the tail of one to the tip of the other (if you want a visual reference)
  • Add the x components of each vector together to get the x component of your resulting vector
  • Add the y components of each vector together to get the y component of your resulting vector 
  • Write your x and y results in rectangular form as a new vector 

Try this interactive tool!

Adjust the start and end points of vectors \( \vec{u} \), \( \vec{v} \), and the resultant vector \( \vec{u} + \vec{v}\). Use the checkboxes to toggle the visibility of each vector.

Practice

To practice, try adding the following vectors: 

1) \( \vec{v}_{\,1} = (3,-1) \) and \( \vec{v}_{\,2} = (4,1) \)

2) \( \vec{v}_{\,1} = (14,10) \) and \( \vec{v}_{\,2} = (-3,-3) \)

3) \( \vec{v}_{\,1} = (-3,-3) \) and \( \vec{v}_{\,2} = (4,-3) \)

 

 

Answers:

1) \( \vec{v}_{\,result} = (7,0) \)

2) \( \vec{v}_{\,result} = (11,7) \)

3) \( \vec{v}_{\,result} = (1,-6) \)

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