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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 Subtraction

Vector subtraction is similar to addition except it is often used to find one of the original vectors from the resultant vector.

For example, if we have \( \vec{v} _{\,1} + \vec{v} _{\,2} = \vec{v} _{\,r} \) but we're only given \( \vec{v} _{\,1} \) and \( \vec{v} _{\,r} \), we can find \( \vec{v} _{\,2} \) by solving our previous equation for \( \vec{v} _{\,2} \) like this: \( \vec{v} _{\,2} = \vec{v} _{\,r} - \vec{v} _{\,1} \). To subtract \( \vec{v} _{\,1} \) from \( \vec{v} _{\,r} \), we need to add the negative of \( \vec{v} _{\,1} \) like this:

\( \vec{v} _{\,2} = \vec{v} _{\,r} - \vec{v} _{\,1} \)

        \( \vec{v} _{\,2} = \vec{v} _{\,r} + ( -\vec{v} _{\,1} ) \) 

This is useful for when we want to remove the effects of vectors from others. For example, if a boat is travelling in the same direction as the wind and we want to know how fast the boat would be moving without the wind, we can subtract the wind from the boat's speed, or, as we learned before, we can add the negative of the wind.

But what is the negative of a vector?

The length of the vector will always be a positive quantity, so the negative of a vector will only change the direction. Specifically, it will point in exactly the opposite direction of the original vector.

For example, if we have a vector in rectangular coordinates like \( \vec{v} = (3,-4) \), we can get it to point in the opposite direction by changing the sign of both the x and y components. If we do that, we would have \( -\vec{v} = (-3,4) \). As you can see, the vector is reflected through the x and y axes, so the direction is completely flipped, but the length is still the same.

Plane Example

A good example of vector subtraction and vector relativity is with airplanes. Because airplanes generate their lift based on how fast the air moves over the wing, planes need to measure their speed relative to the ground and relative to the air, like we can see in the following problem:

 

Example: A plane is moving north at 200km/h (relative to the ground) and the wind is moving north at 30km/h (relative to the ground). How fast is the plane moving relative tot he air?

Solution:

In this problem, we can find how fast the plane is moving relative to the air by subtracting the wind's speed from the plane's speed. By doing this, we are subtracting the effect of the wind from the system so the wind is no longer moving. This means we can treat the wind as our reference like the ground was before.

1. First, we can label the plane speed as \( \vec{v} _{\,plane} \) and the wind speed as \( \vec{v} _{\,wind} \). Next, we can plot the vectors to get an idea of what they look like and what our plane speed relative to the wind, \( \vec{v} _{\,r} \), will look like. If we use north as the +y direction (like on a compass), we will have: \( \vec{v} _{\,plane} \) \( = (0,200) \) and \( \vec{v} _{\,wind} \) \( = (0,30) \) whch can be graphed like this:

Note that \( \vec{v} _{\,wind} \) is shifted slightly to the right to make it easier to see. It is actually on top of \( \vec{v} _{\,plane} \).

2. Next, we need to subtract the wind velocity from the plane velocity to find the velocity of the plane relative to the wind, \( \vec{v} _{\,r} \), so that we can use the wind as our reference, as mentioned before. We perform this subtraction like so:

\( \vec{v} _{\,r} \) \( = \) \( \vec{v} _{\,plane} \) \( - \) \( \vec{v} _{\,wind} \)

       \( \vec{v} _{\,r} \) \( = \) \( \vec{v} _{\,plane} \) \( + \) ( \( -\vec{v} _{\,wind} \) )

To get \( -\vec{v} _{\,wind} \) we need to change the sign of the x and y coordinates of \( \vec{v} _{\,wind} \). This would give us \( -\vec{v} _{\,wind} \) \( = (0,-30) \).

 

3. Now, as we did with vector addition, we need to add \( \vec{v} _{\,plane} \) and \( -\vec{v} _{\,wind} \) tip to tail, which would look like this:

Note that \( -\vec{v} _{\,wind} \) and \( \vec{v} _{\,r} \) are shifted slightly to the right to make them easier to see. They are actually on top of \( \vec{v} _{\,plane} \).

 

4. Now we need to add the x and y components of \( \vec{v} _{\,plane} \) and \( -\vec{v} _{\,wind} \) to get the x and y components of \( \vec{v} _{\,r} \) which we'll label \( v _{\,r x} \) and \( v _{\,r y} \).

So for \( v _{\,r x} \) we would have:

\( v _{\,r x} \) \( = \) \( v _{\,plane} \) \( _{\,x} \) \( + \) ( \( -v _{\,wind} \) \( _{\,x} \) )

\( v _{\,r x} \) \( = 0 + 0 \)                             

\( v _{\,r x} \) \( = 0 \)                                    

Similarly, for \( v _{\,r y} \):

\( v _{\,r y} \) \( = \) \( v _{\,plane} \) \( _{\,y} \) \( + \) ( \( -v _{\,wind} \) \( _{\,y} \) )

\( v _{\,r y} \) \( = 200 + (-30) \)                

\( v _{\,r y} \) \( = 170 \)                                

 

6. So the velocity of the plane relative to the wind is \( \vec{v} _{\,r} \) \( = ( \) \( v _{\,r x} \), \( v _{\,r y} \) \( ) = (0, 170) \).

 

If the plane being slower relative to the wind than the ground is unintuitive to you, you can think about it like when you pass by a car on the highway. That car appears to be moving slowly relative to you because your car and the other car are both traveling in the same direction. In this case, both the plane and the wind are traveling in the same direction.

Subtraction Summary

In summary, to subtract vectors in rectangular coordinates we need to:

  • Graph the original vectors and figure out which one is being subtracted from which
  • Find the negative of the vector being subtracted by changing the sign on the x and y components
  • 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. You can also choose to use the positive vector \( \vec{v} \), or the negative vector \( - \vec{v} \).

Practice

To practice, try subtracting the following vectors:

1) \( \vec{v} _{\,1} = (5,-2) \) and \( \vec{v} _{\,2} = (-1,-1) \). Subtract \( \vec{v} _{\,2} \) from \( \vec{v} _{\,1} \).

2) \( \vec{v} _{\,1} = (-7,4) \) and \( \vec{v} _{\,2} = (3,-3) \). Subtract \( \vec{v} _{\,1} \) from \( \vec{v} _{\,2} \).

3) \( \vec{v} _{\,1} = (13,3) \) and \( \vec{v} _{\,2} = (5,7) \). Subtract \( \vec{v} _{\,2} \) from \( \vec{v} _{\,1} \).

 

 

 

Answers:

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

2) \( \vec{v} _{\,result} = (10,-7) \)

3) \( \vec{v} _{\,result} = (8,-4) \)

Another Example

If you would like another example, take a look at the video below:

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