If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Algebra (all content)

### Course: Algebra (all content)>Unit 20

Lesson 16: Solving equations with inverse matrices

# Matrix word problem: vector combination

Sal finds the appropriate combination of two given vectors in order to obtain a third given vector. This is done by representing the problem with a single matrix equation and solving that equation. Created by Sal Khan.

## Want to join the conversation?

• Can anyone point me to a video of vectors introduction? I would like to review basic operation and concepts. Something like what Khan starts doing at • Hi joaocunhajeronimo.

You can watch the introduction to vectors and scalars in the Physics list: http://www.khanacademy.org/video/introduction-to-vectors-and-scalars?playlist=Physics

I can’t recommend you other videos about vectors because I didn’t see the physics list yet, but I hope you can find them. You can also read a little bit on Wikipedia: http://en.wikipedia.org/wiki/Euclidean_vector (I recommend you the “Representations” section)

Also you should remember that the matrices are arrangements of numbers that represent something else; in this case it represents a vector. When you have a matrix with just one column it is also called a column vector; analogously happens the same with matrices whit just one row (row vector). In this case, the first number represents the “x” value and the second represent the “y” value. If you have a 3x1 matrix (or vector) or a 1x3 matrix (or vector), the third value represents the “z” value. When you have more than 3 values the thing gets funnier because it can represent time or temperature or anything else but I don’t want to confuse you.

Sorry for my poor english, not my native language. Hope this help you, see you.
• Since order matters with matrix multiplication, why is Sal multiplying the adjoint of vector A with vector [7, 6] before multiplying the reciprocal of A's determinant with it's adjoint? Is this allowed by some property of matrix inversions? I tried multiplying through in both orders, and got the same result; why does this work in this case? • what is the difference between linear algebra and original algebra • I am dealing with something called a Tensor, but for me it is just like a matrix, in fact is is a matrix as far as operations and math goes, and for what we use them in class. But my teacher and books of engineering keep calling them tensors, and the definitions I found in internet are quite useless, they keep using the word tensor and tensor field to define tensor... So can anyone give me an easy to go and understand explanation of what a tensor is, and why a tensor of second order can be viewed as a matrix, and what would be a tensor of third order and the difference with a matrix... Is it really helpful the definition of tensor or it is just another jargon or nomenclature more in the world of mathematics... • At , Sal says that you have to draw the vector so that the orientation and magnitude are correct. What are orientation and magnitude? • The 'magnitude' means the size. It's the length of the vector. The 'orientation' just means direction. That's the way the vector is pointed. Two vectors are equal if they have the same length, and they point in the same direction. They can be drawn anywhere on the graph because vectors only convey "magnitude" and "direction," NOT a starting point.

So, for example, a vector whose tail is at (1,1) and whose head is at (4,5) is the same as a vector whose tail is at (2,4) and whose head is at (5,8). They both have the same length. They are both 5 units long (from the pythagorean theorem... they both go up 3 units and to the right 4 units. So the square root of 3 squared plus 4 squared is 5). Also they both point the same direction, because they both have a slope of 3/4. Therefore, even though they are not in the same place on the graph, they are the same vector because their lengths and their directions are the same.
• Why would you want to find how to add two vectors to get a third? • One application is physics when you want to find the direction of a force.

Imagine two horses pulling a wagon. Let's say the gray horse is pulling the wagon with a certain force which we will call G. (The standard unit of force is the Newton, but we'll just use letters for now) The brown horse (which is a bit stronger) is pulling with force B. The total force pulling the wagon forward is G + B.

However, and here is where it gets interesting: Suppose that we now attach the gray horse to the back of the wagon and have him pulling in the opposite direction. You have two forces acting on the wagon in opposite directions. B pulling forward and G pulling backward. Now there is still G and B force acting on the wagon, but they are acting against each other. The resultant force pulling the wagon forward is B-G.

Now, if we had two of the Brown horses, one attached to each end of the wagon, the resultant force would be B-B or 0. So, even though 2B of force is acting on the wagon, since each B is acting in the opposite direction, it is as though no force were acting on the wagon and it does not move. Think of that as adding Vector B to Vector -B and getting 0.

Now imagine that two brown horses are attached to the front of the wagon but one is pulling 45 degrees to the right and one is pulling 45 degrees to the left. What direction will the wagon go? Without getting into the equations, we can nevertheless see that the wagon will go forward because each horse is pulling at the same angle, but the resultant force on the wagon will not be 2B. It will be less, because the horses are pulling in different directions. And of course, if one horse decided to pull at 50 degrees, the wagon would start to veer in that direction. Or, if the weaker gray horse pulled right at 45 degrees with a stronger brown horse pulling left at 45 degrees, the wagon would veer left.

Now, think of the horses and their force as vectors. Vector G + Vector B = Vector W where G is the grey horse, B is the brown horse and W is the resultant force and its direction acting on the wagon.

I hope that example helps and makes sense.
• So, when I read about vectors in my geometry textbook and they say the notation for writing a vector is〈x,y〉it's the same as when Sal writes them as column vectors? (I suppose writing them as column vectors makes it easier to solve the problem.)
(1 vote) • At 12.53, Sal uses a word "vanier reality", can anyone please tell me what that means? I loved the way he connected philosophy and matrices. It would be nice if someone could tell me how to spell the word "vanier"? My english vocabulary is pretty weak and I wish to improve it. • At , what is the origin of the word "isomorphism"?  