Amy has a master's degree in secondary education and has taught math at a public charter high school.
After watching this video lesson, you should be able to add, subtract, and multiply your vectors. Learn how easy it is to perform these operations and what you need to keep in mind when performing these operations.
All right, let's go!
We talk about vectors in this video lesson. What are they? We define them as measurements that include both magnitude and direction. Just think of an arrow. We can draw arrows of different lengths, but each arrow has its own direction. This is exactly what a vector is.
And guess what? We draw them on our Cartesian plane; they look just like our arrows. Remember how we write our points on the Cartesian plane with parentheses and then our values separated by commas? Our points look like (3, 4) with our x value first and then our y value (x, y).
We write our vectors similarly, as well. When we draw our vectors on our Cartesian plane, we usually have the vector beginning at the point (0, 0) and we mark the point where the vector ends. We mark the end point the same way we mark our Cartesian points with our x value first and the y value second. So, we label a vector that starts at the point (0, 0) and ends at the point (5, 6) with the notation (5, 6) to let us know that it has an x length of 5 and a y length of 6.
Most times you will see vectors identified with just two values, but sometimes you will see vectors being identified with more than two values. Theoretically, vectors can be identified with as many values as needed. We could have a vector identified with 3 or even 5 values. For example, (x, y, z) or even (1, 2, 3, 4, 5).
Using this notation for our vectors, we can easily perform our operations of addition, subtraction, and multiplication. These vector operations become very easy when we break the notation apart into the vector's separate parts. Even though our vectors may have two or more numbers that identify them, when we do the operations, we do it in stages. First, we add, subtract, or multiply just the x values. Then we add, subtract, or multiply just the y values, and so on and so forth until we've covered all our identifying values.
Let's look at addition.
Add vector A (1, 2) and vector B (5, 8).
To add these two vectors, we separate each vector into its x part and its y part. For vector A, our x is 1 and our y is 2. For vector B, our x is 5 and our y is 8. Adding them, we get A + B = (1 + 5, 2 + 8) = (6, 10). Do you see how we simply added the two vectors in parts? That's all we have to do. The addition part is the same as we've always done it.
Subtraction is also very similar. We break the two vectors into their separate parts and then perform a subtraction on each of the parts.
Subtract vector A (5, 6) from vector B (11, 23).
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We have to be careful when we read problems versus when we see math problems written. When we read this problem, it tells us that we are subtracting A from B, so mathematically, we write B - A. We have to be careful not to write A - B because that will give us a completely different answer. When we see problems written out mathematically already, we can rest assured that everything is in the correct order.
So our problem is B - A. Let's break our vectors into their separate parts. Vector A has an x part equal to 5 and a y part equal to 6. Vector B has an x part equal to 11 and a y part equal to 23. Performing the subtraction, we get B - A = (11 - 5, 23 - 6) = (6, 17). We took each part and we performed the appropriate subtraction to each part. What do you think? Not too bad? Just remember, break the vector into its separate parts and then do the operation like you always do.
The same goes for multiplication.
Multiply the vector A (-2, 3) with 4.
We are multiplying our vector with a number. Again, we break our vector into its parts and then perform the multiplication to each part. We get 4*A = (-2*4, 3*4) = (-8, 12). We multiplied each part by 4 to get our answer. And that's all we had to do!
Let's review what we've learned:
We learned that vectors are measurements that include both magnitude and direction. If our vector begins at the point (0, 0), then we label our vector with its end point coordinates on the Cartesian plane. For example, the vector (7, 4) ends at the point (7, 4). This means that this vector has an x length of 7 and a y length of 4. It is pointing away from the origin towards the end point. To add, subtract, and multiply vectors, we separate the vectors into separate parts and then perform the operation to each part.
After reviewing this lesson, you should have the ability to:
Identify the notation for vectors
Explain how to add, subtract, and multiply vectors
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