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.

- Welcome
- Learning Math Strategies (Online)Toggle Dropdown
- Study Skills for MathToggle Dropdown
- Business MathToggle Dropdown
- Place Value in Decimal Number Systems
- Arithmetic Operations
- Basic Laws
- Operations on Signed numbers
- Order of Operations
- Some Useful Basic Numeracy
- Fractions
- Decimals
- Percents
- Ratios and Proportions
- Exponents
- Statistics
- Factoring
- Rearranging Formulas
- Solving Linear Equations
- Solving Systems of Linear Equations
- Trade and Cash Discounts
- Multiple Rates of Discount
- Payment Terms and Cash Discounts
- Markup
- Markdown
- Simple Interest
- Equivalent Values
- Compound Interest
- Equivalent Values in Compound Interest
- Nominal and Effective Interest Rates
- Simple Ordinary Annuities
- Simple General Annuities

- Hospitality MathToggle Dropdown
- Place Value in Decimal Number Systems
- Arithmetic Operations
- Order of Operations
- Basic Laws
- Prime Factorisation and Least Common Multiple
- Fractions
- Decimals
- Percents
- Exponents
- Units of Measures
- Fluid Ounces and Ounces
- Metric Measures
- Yield Percent
- Recipe Size Conversion
- Ingredient Ratios
- Food-Service Industry Costs

- Engineering MathToggle Dropdown
- Basic Laws
- Order of Operations
- Prime Factorisation and Least Common Multiple
- Fractions
- Exponents
- Radicals
- Reducing Radicals
- Factoring
- Rearranging Formulas
- Solving Linear Equations
- Areas and Volumes of Figures
- Congruence and Similarity
- Functions
- Domain and Range of Functions
- Basics of Graphing
- Transformations
- Graphing Linear Functions
- Graphing Quadratic Functions
- Solving Systems of Linear Equations
- Solving Quadratic Equations
- Solving Higher Degree Equations
- Trigonometry
- Graphing Trigonometric Functions
- Graphing Circles and Ellipses
- Exponential and Logarithmic Functions
- Complex Numbers
- Number Bases in Computer Arithmetic
- Linear Algebra
- Calculus
- Set Theory
- Modular Numbers and Cryptography
- Statistics
- Problem Solving Strategies

- Upgrading / Pre-HealthToggle Dropdown
- Basic Laws
- Place Value in Decimal Number Systems
- Decimals
- Significant Digits
- Prime Factorisation and Least Common Multiple
- Fractions
- Percents
- Ratios and Proportions
- Exponents
- Radicals
- Reducing Radicals
- Metric Conversions
- Factoring
- Solving Linear Equations
- Solving Quadratic Equations
- Functions
- Domain and Range of Functions
- Polynomial Long Division
- Exponential and Logarithmic Functions
- Statistics

- Nursing MathToggle Dropdown
- Arithmetic Operations
- Order of Operations
- Place Value in Decimal Number Systems
- Decimals
- Fractions
- Percents
- Ratios and Proportions
- Interpreting Drug Orders
- Oral Dosages
- Dosage Based on Size of the Patient
- Parenteral Dosages
- Intravenous (IV) Administration
- Infusion Rates for Intravenous Piggyback (IVPB) Bag
- General Dosage Rounding Rules

- Transportation MathToggle Dropdown
- Physics

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.

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:

**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 \( \vec{v} _{\,wind} \) \( = (0,30) \) which 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.

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 \( - \vec{v} \).

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

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

- Last Updated: May 18, 2023 3:19 PM
- URL: https://libraryguides.centennialcollege.ca/mathhelp
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