Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
4
Academic year
2022
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77
Writing equations as dot products Here is the question: find a vector v that is perpendicular to 1, comma, one-half, (1, 1/2). Find v perp to (1,12) The answer is we can check if it's perpendicular by taking the dot product. We want the dotproduct of v1 and v2 to be 0. If we write out this dot product, we get v1 plus one-half of v2. 0=v*1,12=v1+12v2 So, there are many possible solutions to this problem. If you pick any number v1, then youcan figure out what v2 has to be to get this to add up to zero. Here are a couple. For example, v1 could be negative 1, and then v2 would have to be 2to get this thing to add up to 0. Or if v1 were negative one-half, then this would have toequal 1 to get the sum to be 0. -1,2 or (-12,1) Pretty simple. Let's try to visualize the vectors to check that they are working as expected. It is better to draw labeled vectors so that these lengths are equal so the angles look right. The vector is here, 1, a half. And then, negative 1 is here. Thus, here is the vectornegative 1, comma, 2. Look logically. Here is negative one-half, 1 drawn.
These two vectors share the same direction. That is reasonable because they're both perpendicular to one, comma, a half. That is whythey are going in the same direction. We can see from the picture that there should be a lot of them, cause you are able to takeany vector in this direction, and it'll be perpendicular to 1, comma, 1/2. Last time these alsowere in the same direction. Because this vector is 2 times that vector. It all makes sense. Thus, there are many vectors that are perpendicular to this one and they all go along thisline. Now, let us examine the question, "How do we find a vector that is perpendicular to aline?" Here is our new question. Given the line x + ½ y = 0, find a vector perpendicular to it. x+12y=0 Thus, to solve this problem, it is helpful to look for a hidden dot product. A dot product is familiar to us. It is common in geometry that to write some expressions asdot products. Dot products are notated as a scalar times a vector, but we could write this one as ahidden dot product. So x plus 1/2 y, it can be written as (x, comma, y) dotted with (1, comma, 1/2). It turns out that writing it that way is amazing, it is helpful with understanding the meaning. Remembering that, draw an image of this line and draw in this vector. this vector is goingto be a main thing. x-axis, y-axis. Thus, the line is: x + ½ y = 0. And we draw in this vector with a magnitude ofone-half because it comes up in that formula. It will help us to have it in the picture, so to speak, to think about what everything means. Here it is the vector one, comma, a half.
Let's suppose we have a point (x, y) on the line; then perhaps its coordinates are (x,comma, y). So, if x and y satisfy this equation, then 0 equals x plus one-half y. That's what does it mean for (x, y) to be on the line. We will now see if this gives us an insight into how to write x + one half y as a dot product.Let’s go. So it is (x, y) dotted with (one, comma, one-half). Now, this dot product is 0. This tells us that the vector (x,y) is perpendicular to the vector(1, comma, a half). Let's draw the vector (x, y) in our picture. It's the vector that goes from the origin to (x, y).Here it is. This equation shows that the vector is perpendicular to that vector. So there is a rightangle here.
We see that the vector (1, comma, one-half) is perpendicular to this line.
Writing Equations as Dot Products
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