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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.

**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: \( \vec{v}_{\,result} \) \( = \) (\( \vec{v}_{\,boat} \),\( \vec{v}_{\,wind} \)) \( = \) (\( 40km/h \),\( 10km/h \))

**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:

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

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

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.

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

- Last Updated: Aug 1, 2024 10:46 AM
- URL: https://libraryguides.centennialcollege.ca/mathhelp
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