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 will be able to find the dot product of vectors both algebraically and geometrically. Learn the difference between the two and what you need in order to calculate them.
Imagine holding two arrows with their flat ends touching. What do they look like? Since we are talking about math, they look like two vectors. See, a vector is a measurement with both magnitude and direction. Hence, we have a particular length and an arrow telling us which direction our vector is in. So an arrow is a perfect real-world representation of a vector. Just think about it. When you hold an arrow in your hand, no matter which direction you point it in, it is always the same length and same magnitude long, but the direction changes depending on which way you point the arrow. It is the same with a vector.
In math, we can draw our vector on our Cartesian coordinate plane. We can note the vector two ways. We can note it with its starting point and its end point on the Cartesian plane.
Or we can note it with its magnitude and direction.
When we have two vectors, we can note both with their start and end points.
Or we can note the magnitudes along with the angle between the two vectors.
Since we are not always given the magnitude, we need a way to find the magnitude of a vector if we are just given the beginning and end points. To do this, we use this formula:
This formula is telling us that for a vector A, the magnitude is found by squaring the x length and the y length, adding them together, and then taking the square root. If you notice in our vectors so far, I've shown them starting at the point (0, 0). If they start at a different point, we'll need to calculate the x-length and the y-length before using the magnitude formula. Alternatively, we can move the vector so that its starting point is at (0, 0) and then use the new end point to calculate our magnitude.
Let's look at how we use this formula. Let's say we have an arrow that has its flat end at the point (0, 0) and is pointing to the point (3, -4). To calculate its magnitude, we plug in 3 for x and -4 for y. We get the sqrt.(3^2 + (-4)^2). This evaluates to sqrt.(9 + 16) = sqrt.(25). Taking the square root of this, we get 5. So the length or magnitude of this arrow is 5.
Notice that we kept the negative sign since the arrow is pointing downwards. Because vectors have direction, it is important to keep any positive or negative signs that we have.
Now let's talk about the dot product. This is the multiplication of two vectors. We get a scalar result, meaning we get a simple number instead of a number with direction. We have two formulas we can use to find the dot product depending on whether we are given beginning and end points or we are given magnitudes with an angle.
The formula to use when we are given beginning and end points is this one:
So, what we do is we take the x lengths of each vector and multiply them together; then we take the y lengths of each vector and multiply those together. Then we add them up to find our dot product.
The formula to use when we are given magnitudes and an angle is this one:
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This formula is telling us to multiply the two magnitudes together, then find the cosine of the angle between them. We then multiply it all together to find our dot product.
Even though these two formulas are completely different, you will find that they will give you the same answer for the same situation.
Let's take a look, shall we?
Let's use the first formula for the beginning and end points. We will call the vector on the right vector A and the vector on the left vector B. So for vector A, the x length is 3 and the y length is 4. For our vector B, the x length is -6 and the y length is 8. Plugging these into our formula, we get A*B = 3*-6 + 4*8. This evaluates to A*B = -18 + 32 = 14. So our dot product from this formula is 14.
Let's try the other formula. Our magnitude for vector A is 5 and our magnitude for vector B is 10. The angle between them is 73.7 degrees. Plugging these into our formula we get A*B = 5*10*cos (73.7). This evaluates to A*B = 50* 0.2806667 = 14, approximately.
They both give us the same answer, except that for the second formula we just needed to do a little bit of rounding.
So, what did we learn?
We learned that a vector is a measurement with both magnitude and direction. We draw vectors on our Cartesian coordinate plane. We can note vectors with either beginning and end points or their magnitudes with direction. The dot product is the multiplication of two vectors. The formula to use when we are given beginning and end points is this one:
We multiply the x lengths of each vector and the y length of each vector together and then add them up.
The formula to use when we are given the magnitudes and the angle between the vectors is this one:
We multiply the magnitudes together, take the cosine of the angle, and multiply it all together.
The result of the dot product is a scalar number without direction.
This lesson will help you gain the skills needed to:
Recognize a vector
Draw vectors on the Cartesian plane
Apply the formula for finding the magnitude of vectors
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