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What is the magnitude of the vector π΄π΅, where π΄ is the point with coordinates five, negative nine and π΅ is the point with coordinates nine, one?
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First, letβs recall how to calculate the magnitude of a vector.
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If a vector π£ is in component form with components π₯ and π¦, then the magnitude of this vector, which is denoted using the vertical bars on either side, is found by calculating the square root of π₯ squared plus π¦ squared.
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This is just an application of Pythagorean theorem as the vector forms a right-angled triangle with its π₯- and π¦-components.
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So the first step to answering this problem is we need to write the vector π΄π΅ in component form.
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The π₯-component of the vector π΄π΅ is found by subtracting five from nine.
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The π¦-component is found by subtracting negative nine from one.
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Therefore, in component form, the vector π΄π΅ is four, 10.
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So now we can substitute into our formula for calculating the magnitude of a vector.
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The magnitude of π΄π΅ is the the square root of four squared plus 10 squared.
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Four squared is 16 and 10 squared is 100, so the magnitude of π΄π΅ is the square root of 116.
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Now this surd can be simplified if we recall that 116 is equal to four multiplied by 29.
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The laws of surds tell us that we can separate this out into the the square root of four multiplied by the square root of 29.
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Four is a square number, so we can find its square root exactly; itβs just two.
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29 is not a square number and, in fact, itβs a prime number, so we canβt simplify this surd anymore.
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Therefore the magnitude of the vector π΄π΅ is two root 29.