WEBVTT
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The vector 𝐯 is shown on the grid of unit squares below.
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Find the value of the magnitude of 𝐯.
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In the process of reading out the problem, I’ve given away the fact that this piece of notation here the vector 𝐯 in between the two vertical lines represents the magnitude of 𝐯.
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One way of thinking of vector quantities is as quantities which have both magnitude and direction.
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For example, if I tell you that a certain town is three miles northeast of where I am, that’s a vector information because I’m telling you not only how far away it is, but in what direction.
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You might know that three miles northeast is a displacement.
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The magnitude of that displacement is just the distance, three miles.
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So we throw away that direction information.
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Have a look at the vector in the diagram, this is an abstract geometric vector, which could represent a displacement or a velocity or a force or something else.
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For a geometric vector like the one we have, the magnitude is just the length.
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How do we find the magnitude or length of this vector?
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We use the grid of unit squares that it is drawn on.
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We draw a right triangle, the base of which lies on top of four unit squares and so has a length of four units.
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And in a similar way, we see that this side has a length of three units.
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The magnitude of 𝐯 is then the length of the hypotenuse, which we can find using the Pythagorean theorem.
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Applying the Pythagorean theorem, we get that the length of the vector is the square root of three squared plus four squared.
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Three squared plus four squared is 25.
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And so the length is the square root of 25, which is five.
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Remember this is not only the length of the hypotenuse in the diagram.
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This is the magnitude of the vector that we wanted to find.
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To specify our vector exactly, we need not only this magnitude five, which tells us how long the vector is, but also the direction in which the vector is pointing.
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Vectors have both a magnitude and a direction.
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And in this problem, we’ve seen how to calculate the magnitude when the vector is represented geometrically in the plane.