Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
4
Academic year
2022
Sporkz
Views
24
Scalar multiplication ● Scalar multiplication of vector is a process of multiplying a vector by a scalar. A scalar is a number that does not depend on any other quantity in the equation. ● Scalar multiplication does not change the direction, just the length of a vector.● Lambda of a vector is the scalar value that corresponds to its length. It is represented by the Greek letter "λ." We can also scale vectors. We can stretch them or contract them. If we write something like 2v, it means that the vector has the same direction as v but is 2times longer. Let’s visualize. Suppose that was v. What would 2v look like? It would be in the same direction as v, buttwice as long. And that's 2v. What would be the components of that [v1 , v2]? Well, the first component of distance ishow far it is. The second component is how high it is. We can determine those ratios by using similar triangles. The small triangle is similar tothe big triangle. The hypotenuse of the big triangle is twice as long as the side of the smalltriangle. 2v is twice as long as v. Therefore, all of the sides should be twice as long. If thecorresponding side of the big triangle is 2 times v1, then the side of a small rectangle withthe same perimeter is 2 times v1.
Thus, the components of 2v are 2 times v1 and 2 times v2. Similarly, a negative vector has a direction opposite to that of vector v and the samelength. So negative v looks like that, while positive v is the opposite direction. Thus, to sum up, if v is in the same direction as w, this means that v is lambda times wwith lambda being a positive number. Any vector that has the same direction as v, say v1 from here to here, is lambda times v.And for the one I'm indicating, lambda would be about 1.5. If v goes in the opposite direction, then v should be lambda times w, where lambda is anegative number. Thus, for example, negative v going in the opposite direction of v, or negative 2v, is goingin the opposite direction and twice as long. Now I’ll show you one moment that can be tricky. So is the vector [2, 3] in the samedirection as the vector [4, 7]? And how can we determine that? If we see in the same direction, we are likely to think about this. Since the vectors were inthe same direction, this meant that v was lambda times w. What we want to know iswhether [4, 7] is equal to lambda times [2, 3] for some number of lambda. If we solve the equation for lambda, then they are in the same direction. And if there is no lambda, if we figure out that no solution exists, it means the two lines arenot in the same direction. Now we have lambda times [2,3] is [2 lambda , 3 lambda], exactly like 2v is to [2 v1 , 2 v2]. What is the meaning of two vectors being equal to each other?
If you write the components of an equation in this way, it means that the first componentsare equal and the second components are equal. We're trying to say that 4 equals 2 times lambda. We'd also want 7 to be 3 times lambda. If we attempt to solve the equation, we will find that the first thing tells us lambda is 2, butthe second thing tells us lambda is 7/3. As a result, all the numbers are different, and there is no constant (lambda) that can beused in the equation to make everything work. And since the vectors for these two functions are not in the same direction, there is nolambda. Here are pictures for us to become even more sure. Their directions do not match. In that picture, the black vector is 2 comma 3 and the dotted factor is 4, 7. Although they appear to be moving in the same general direction, the two lines are notexactly parallel. In the diagram on the right, I have added one more red vector, which points in the samedirection as 2 and 3. That vector is 4 and 6. So let's put it down here. Thus, 4 times 6 is 2 times 3. This is in the same direction as 2 times 3.
It is the red vector in the picture. The lesson to be learned from this is that whenever we see a question about vectors thatinvolves two things being in the same direction, then algebraically, we can work with that.
Scalar Multiplication
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