Vector Equations Simplified (Step-by-Step Tutorial)

To truly master linear algebra, you must fully grasp the power of vectors because vectors are the key to understanding matrices.

Jenn (B.S., M.Ed.) of Calcworkshop® teaching vector equations

Jenn, Founder Calcworkshop^®, 15+ Years Experience (Licensed & Certified Teacher)

Why?

Because a vector is actually a matrix in disguise!

Defining Vectors and Vector Equations

So, what are vectors or vector equations?

A vector is a quantity with both magnitude (length) and direction. If I tell you to “walk 3 miles,” this only tells you the magnitude or length I wish you to walk. But if I say, “Walk 3 miles north by northwest,” then I’m giving you both a magnitude and direction- a vector!

Vector Notation and Operations

A vector can be denoted using a bold typeface

u

or with an arrow

\vec{u}

And two vectors are equivalent if they have the same size and direction and only differ in position (i.e., $u = v)$ .

Equivalent Vectors (Same Length & Direction)

Vector Addition and Scalar Multiplication

Multiplying a vector by a scalar (number) will change the magnitude but not the direction.

Vector Scalar Multiplication

Multiplying by a negative scalar will cause the vector to point in the opposite direction.

Vector – Negative Scalar Multiplication

When we add to vectors, called vector addition, it forms the diagonal of a parallelogram.

Vector Addition – Parallelogram Method

Okay, so all of this should make sense to us, or at least dust off the cobwebs from when we first learned about vectors from precalculus.

But now it’s time to talk about vector equations, as they are any function that takes on one or more variables and returns a vector.

The best way to make sense of this is, to begin with an ordered pair in $R^{2}$ . Now, $R^{2}$ means the set of all vectors with two entries. But more simply, $R^{2}$ refers to the xy-plane or the Cartesian plane.

Vector Spaces and Column Vectors

But here’s what is so cool… any coordinate $(x, y)$ can be written as a vector, sometimes called a column vector, $(\begin{array}{l} x \\ y \end{array})$ in the plane $R^{2}$ !

For example, the following are two-entry vectors. Notice that each has an initial point at the origin and a terminal point at the ordered pair.

Graphing Vectors – Example

But there’s more to life than just two-entry vectors! Let’s look at $R^{3}$ , which is the set of all ordered triples of real numbers and is also called 3-space (i.e., 3-D vector).

For example, the ordered triple in three-space has an initial point at the origin and terminal point $(2, 4, 5)$ .

Plotting a Vector in Three Space

Extending Vectors to Multidimensional Space

While we can only graph vectors in $R^{2}$ (i.e., $2 \times 1$ column matrices) and $R^{3}$ (i.e., $3 \times 1$ column matrices) we can carry out calculations, analysis, and arithmetic for n-tuples of real numbers in multidimensional space that we denote as $R^{n}$ , pronounced “r-n” for $n \times 1$ column matrices.

Basic Properties of Vectors

Additionally, there are some basic properties of vectors that are important for all vectors $u, v, w$ in $R^{n}$ and all scalars c and $d$ :

$\vec{u} + \vec{v} = \vec{v} + \vec{u}$
$(\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})$
$\vec{u} + \vec{0} = \vec{0} + \vec{u} = \vec{u}$
$\vec{u} + (- \vec{u}) = (- \vec{u}) + \vec{u} = \vec{0}$
$c (\vec{u} + \vec{v}) = c \vec{u} + c \vec{v}$
$(c + d) \vec{u} = \vec{u} + d \vec{u}$
$c (d \vec{u}) = (c d) \vec{u}$
$\vec{1 u} = \vec{u}$
$\vec{0} = [\begin{matrix} 0 \\ 0 \\ ⋮ \\ 0 \end{matrix}]$ , where $\vec{0}$ is called the zero vector.

Linear Combination of Vectors

And this brings us to an incredibly important idea: a linear combination of vectors.

Given vectors $v_{1}, v_{2}, \dots, v_{n}$ in $R^{n}$ and scalars $c_{1}, c_{2}, \dots, c_{n}$ , then $c_{1} v_{1} + c_{2} v_{2} + \dots + c_{n} v_{n}$ is called a linear combination of $v_{1}, v_{2}, \dots, v_{n}$ with weights $c_{1}, c_{2}, \dots, c_{n}$ .

Okay, but why is this helpful?

Arithmetic with Column Matrices

Because it allows us to perform arithmetic with column matrices!

For example, suppose

$\begin{aligned} \bar{a} & = [\begin{array}{c} 1 \\ - 3 \end{array}] \\ \bar{b} & = [\begin{matrix} 3 \\ 2 \end{matrix}] \end{aligned}$

Find the following linear combinations:

$\begin{aligned} \bar{a} + \bar{b} & = [\begin{array}{c} 1 \\ - 3 \end{array}] + [\begin{matrix} 3 \\ 2 \end{matrix}] \\ = [\begin{array}{c} 1 + 3 \\ - 3 + 2 \end{array}] \\ = [\begin{array}{c} 4 \\ - 1 \end{array}] \end{aligned}$

$\begin{aligned} 5 \bar{a} + 2 \bar{b} & = 5 [\begin{array}{c} 1 \\ - 3 \end{array}] + 2 [\begin{matrix} 3 \\ 2 \end{matrix}] \\ = [\begin{array}{c} 5 \\ - 15 \end{array}] + [\begin{matrix} 6 \\ 4 \end{matrix}] \\ = [\begin{array}{c} 5 + 6 \\ - 15 + 4 \end{array}] \\ = [\begin{array}{c} 11 \\ - 11 \end{array}] \end{aligned}$

$\begin{aligned} \bar{b} - 2 \bar{a} & = [\begin{array}{c} 3 \\ 2 \end{array}] - 2 [\begin{array}{c} 1 \\ - 3 \end{array}] \\ = [\begin{matrix} 3 \\ 2 \end{matrix}] - [\begin{array}{c} 2 \\ - 6 \end{array}] \\ = [\begin{array}{c} 3 - 2 \\ 2 - (- 6) \end{array}] \\ = [\begin{matrix} 1 \\ 8 \end{matrix}] \end{aligned}$

Isn’t it cool that we can linearly combine vectors just like we do with variables?

Linear Algebra Key Ideas: Spanning Sets

In fact, one of the key ideas in linear algebra is that the collection of all vectors that can be written in the form $c_{1} v_{1} + c_{2} v_{2} + \dots + c_{n} v_{n}$ is the subset of $R^{n}$ spanned (generated) by $v_{1}, v_{2}, \dots, v_{n}$ , denoted as $Span {v_{1}, v_{2}, \dots, v_{n}}$ .

Think of it this way, imagine you want to travel the world. Well, you can “span” the globe using various means of travel: ship, plane, car, train, etc. This collection of options is similar to the spanning set, as they all find ways to formulate a solution or means for our desire to travel.

Next Steps

In this video, you’re going to grasp the concepts of vectors and vector equations, and unravel the relationship between vectors and matrices. Here’s what you can look forward to:

Applying properties of vectors
Performing basic vector operations
Working through examples to identify if a vector is a Linear Combination of another vector
Determining if a vector is within the Span
Writing an equivalent system of equations for a given vector equation and vice versa
Constructing a vector as a linear combination of a set of given vectors
Answering foundational questions about Vector Equations, Linear Combinations, and the Spanning Set

It’s time to dive in and make vectors and vector equations your strength!

Video Tutorial w/ Full Lesson & Detailed Examples

vector-equations-for-matrix-algebra-example

Get access to all the courses and over 450 HD videos with your subscription

Monthly and Yearly Plans Available

Get My Subscription Now

Still wondering if CalcWorkshop is right for you?
Take a Tour and find out how a membership can take the struggle out of learning math.

4.9 / 5

1,136 Reviews

N8Booker11-28-24 - MA, United States

I'm in school to get my degree Networking and Cybersecurity and I'm in my first year. I've been away from school for quite a long time so the concepts that are being taught although not foreign to me, have so much dust on them from a lack of being used in real world applications regularly.I stumbled upon CalcWorkshop by pure chance and it has been a game changer for me. Jen filled in the gaps in between what my Professor is teaching at a rapid fire pace and where my brain grasps and applies it. If not for Jen's approach, I'd have failed my test I took miserably!Am I grateful? It goes beyond words and not I recommend her workshops to everyone!

Jon, C.04-06-25 - MD, United States

CalcWorkshop is a premier resource for any student seeking to understand, practice, and succeed within the vast areas of mathematics. I am impressed with the quality of instruction and practice problems offered by Jenn. As a engineering student, I always utilize Calcworkshop as a the "first stop" to really understand and gain confidence in the subject. I credit my success in Calculus to Jenn and CalcWorkshop and making me "smarter than the average bear."

Sayida S.04-03-25 - PA, United States

I absolutely love this subscription. At first I was iffy about spending the money, but in the long term it is so worth it. I started watching Jenn almost 3/4ths into my calc 2 class, and I a confidently say I understand the depths to each topic. Before, with my professor I felt that I only understood surface level of each topic. But after the CalcWorkshop videos, I have acronyms, tips, and knowledge I walk away with, that I didn't have before. Genuinely cannot wait to keep using this website for my future endeavors, minoring in Mathematics!

Addie M.04-01-25 - United States

The best resource for improving my math skills and improving my performance in Calculus, as well as through other areas that pertain to other math courses! Jenn is an incredible teacher and I recommend Calcworkshop to anyone who experiences insecurity in math.

Gina K.03-30-25 - WA, United States

My experience was great! I needed help studying for my math class, but I wasn't sure how-until an ad for this popped up on my YouTube. After watching the ad, I was convinced, so I subscribed to CalcWoskop. I'm so happy I did because it helped me so much with my class! I learned a lot that I couldn't grasp from my college lectures, and this was the perfect solution.

Serafimk2203-28-25 - NJ, United States

I had an Amazing experience with discrete math and calculus series. Graduating with highest honors thanks to calc workshop. I'll be back on the site again but this time if I need to go back as reference as well as if I get rusty in my career.

Maria03-26-25 - AB, Canada

I found Jenn's videos near the end of my discrete math class, so I only had a month to go through all of her discrete math lessons to study for my exam. I'm sure I would have aced it if I had more time but I got a pretty good grade considering I thought I would fail.So happy to be done that class!As advertised, Jenn is very thorough and doesn't skip steps, she lets you know exactly how she gets from a to b or she will refer you to other videos where she teaches the techniques she's using more in depth. This was extremely valuable to me, being a mature student taking advanced math while not remembering basic algebra.And it was fun to understand and solve complex problems vs memorize proof structures.

RCJK03-07-25 - ID, United States

I had to take Calculus I and II as accelerated classes while working full time. It was my first time taking this courses and was very overwhelming. Calcworkshop really made learning the subjects easier for me by having all the subjects easy to find and easy to understand explanations. I received a B in both classes which I am extremely happy with doing the classes in double time.

Jacqueline S.03-07-25 - SC, United States

Jenn is awesome!Her breakdown of the topic is beyond what you would expect. She is showing you how to arrive at the answer instead of doing a simple equation. Most people need to know the why it give confidence and structure to the learning process. This is what Jenn is great at, she knows how to deliver the information to a non math mind in a way that instills knowledge that you can forever draw from as you move forward. It's really well thought out. She is a true master teacher. Thanks for all that you do for us! Hugs n Blessings... JS

Jujuana03-05-25 - MO, United States

I love the videos the way each lesson is explained step by step. I love that she has examples that you can go over again and again in order to understand the way algebra works. I was lost because my instructor was flying through each lesson and I was totally lost. I attached my syllabus and they helped me stay focus on what I was working and give me a guide to go back and refresh what I went over in class.

Stacey01-31-25 - AZ, United States

Hello, I highly recommend getting a subscription to Jenn and her Calcworkshop teaching math service. I did a general, "go from bad to good at math", Google search and found out about Calcworkshop through responses to the topic on reddit. I've just started watching her videos and found them to be extremely helpful. I'm going back to college after a long absence and math was never my strong suit. So I'm starting at the very bottom of her available subjects and going from there. Just watching a couple of the videos has taken me from being dragged along not understanding to following along just fine in my math class. I'm moving from the "I'll never get this", to the "I look forward to understanding this", mindset in approaching math. This for me is huge, and even though I have a long way to go, I'm confident I will get there thanks to Jenn. Totally worth it!

Kathy W.01-14-25 - WA, United States

Calcworkshop is an outstanding source to learn math and understanding math concepts. Jenn is an outstanding master teacher. Every lesson is clearly presented making understanding easy to grasp. Practice lessons with accompanying answers are given for each section. I am 82, a retired teacher and lifelong learner. Learning calculus has always been on my bucket list. Calcworkshop is helping me reach my goal.

Cheryl01-14-25 - NY, United States

Calc workshop is the absolutely best resource for math and I have seen almost everything that exists out there. I have even recommended it to my daughter's school so that all the kids could use the videos during the summer months for the subject they plan to study in the fall. They are guaranteed to get a A if they have watched the videos and completed the problems before taking the class. Students who are weak in areas can access lower level classes and those who seek to advance above grade level can do so. The videos and exercises are crystal clear and the excellent presentations raise the bar for teachers everywhere. I have always believed that one great math teacher is important in the education process. The videos, exercises and problems are excellent. It is not possible to have ANY knowledge gaps. I have never seen anything on the market that even begins to compete with it!

Justin N.12-22-24 - OK, United States

Calculus can be a scary subject. I know I was scared. I had been out of any math class for over 23 years. I had taken an 8 week Calculus 1 class and felt like I was overloaded. Going into Calculus 2 I found CALCWORKSHOP.com. I used the training videos and material to really dive back into Calc 1 and prepare for Calc 2. Long story short with the material provided and ample study time, I finished Calc 2 with a 95%. Thanks Jenn!

Shelby12-10-24 - OH, United States

CalcWorkshop was a wonderful resource as someone going onto a college calculus 1 class after not having taken a math course in a year! Jenn always explained things in a logical order that built upon previous lessons, and the way she explained topics was always easy to understand. Calcworkshop saved this semester for me, and I couldn't be more grateful to have found this site. :)

Jordan Q.12-09-24 - AR, United States

Anyone struggling with math at any level would greatly benefit from this product. I signed up for it immediately after getting a D on my first exam in calculus 2 and ended up getting A's and B's on the remaining exams, getting a B overall in the class. Jenn breaks down the topics, without skipping any steps, in a way that makes them very easy to understand. I highly recommend this product!

N8Booker11-28-24 - MA, United States

Jon, C.04-06-25 - MD, United States

Sayida S.04-03-25 - PA, United States

Gina K.03-30-25 - WA, United States

See more reviews on Shopper Approved