Teaching Calculus

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  • Nikolay Brodskiy
  • Alexander Brodsky

Part 01 (Transcript)

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Er ... Let me talk today about projections. So the ...

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So our topic for today is projections.

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And the first appearance of projections is naturally in physics.

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When you look at ...

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When you are trying to solve a problem of, for example, pushing an object ...

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So if you want to push an object ...

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... then you apply force.

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And we know force is a vector.

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And I'm applying that force ...

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... and if I applied that force in one direction I'd fail to push the object.

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If I apply that force in another direction the object moves.

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And that's what I want.

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So the result depends on the direction of my vector of force.

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And from practice we know that the more my force is along the direction ...

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... the better the effect is.

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And if it is perpendicular ...

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... it's just a waste of my energy, waste of my effort.

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So er ...

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There are two things here to consider.

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One is a vector of the possible motion of the object.

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And ...

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Sometimes this vector is describing the only direction possible for the object to move.

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Like in the case of pushing a train.

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It is limited to the train tracks and never goes left or right.

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Er ... in case of pushing this eraser it will only go horizontally.

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Otherwise it will fall. So ...

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One vector describes the possible direction and the other vector ...

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Let's call it u.

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... can be thought of as a force applied to the object.

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And then that force ...

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... results in motion.

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And what you want to measure is ...

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... a part of that force that ...

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... acts on the object moving in the direction we want.

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So this is the ...

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... part of the force that actually pushes the object as we want.

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And ...

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The rest of the force ...

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Well let me make it blue. Er ...

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... is a waste of our effort.

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So our force is decomposed into the sum of two vectors.

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One is highly useful for the purpose.

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The other one is total waste.

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And what we want is to maximize the useful one, minimize the waste.

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Sometimes we cannot make the waste zero.

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Er ... but that's a question of technical limitations.

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So our purpose now is to compute both things.

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And first of all we will compute the useful part.

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So to compute that we will need of course that angle.

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Let's call it alpha.

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And then ...

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The magnitude of this useful part of the force ...

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... is going to be called the component.

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This is the notation that we will use for the whole semester.

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Component of the vector u in the direction of the vector v.

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So that is going to be a number.

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And ...

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The value of that number is going to be ...

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... from this triangle ...

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... the magnitude of u times cosine of alpha.

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Now ...

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From computational point of view that formula is not highly useful.

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Because given two vectors u and v ...

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... it is not easy to find the cosine ...

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... unless you use the dot product and the formula we discussed last time.

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So the idea is to ...

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... use ... well, to compute that quantity using the dot product of u and v.

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And to do that I would multiply and divide ...

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... that quantity by the magnitude of v.

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Which doesn't change the quantity itself but the numerator ...

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... now looks like the dot product.

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And actually it is the dot product.

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So it is dot product of u and v ...

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... divided by magnitude of ...

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... v.

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That's the simplest formula for computing ...

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... the component of the force vector onto the given direction.

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Student: Can we [...].

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Yes, you can do that.

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So if you just start with that formula ...

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... u times cosine alpha and ...

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... replace cosine with u dot v divided by magnitude u times magnitude v ...

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... then u cancels and you arrive at the same formula.

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Well, so basically this is the formula that I suggest to memorize.

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And you will use it a lot this semester.

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Now let's talk a little about that formula.

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First of all that's a number.

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Although the whole thing contains a lot of vectors.

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The numerator is the dot product of two vectors resulting in a number.

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The denominator is the magnitude which is a number again.

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