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If the vector π is equal to three π’ minus five π£, π is equal to ππ’ plus five π£, and the cross product of π and π is equal to 50π€, find the value of π.
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To answer this question, weβre going to need to recall what we mean by the cross product of two vectors.
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Essentially, itβs a way of multiplying two vectors.
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Unlike the dot product, where we get a scalar quantity, when we find the cross product of two vectors, we get a vector.
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And we say that if π is the three-dimensional vector given by π one, π two, π three and π is the vector π one, π two, π three.
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Then the cross product of π and π is equal to the vector π two π three minus π three π two, π three π one minus π one π three, and π one π two minus π two π one.
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Now, this isnβt a particularly easy formula to remember.
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And if youβre struggling, you might want to recall that itβs really just the determinant of a three-by-three matrix.
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So now, we have a formula.
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Letβs define π one, π two, π three and π one, π two, π three.
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π one is the horizontal component for π.
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Itβs three.
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π two is the vertical component.
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Thatβs negative five.
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And thereβs no π€-component.
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So π three is equal to zero.
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π one is equal to π.
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π two is equal to five.
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And once again, π three is equal to zero.
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The first element of the cross product of these two vectors then is π two π three.
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So thatβs negative five times zero minus π three π two, which is zero multiplied by five.
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The second element is π three π one, which is zero times π, minus π one π three, which is three times zero.
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And the third element is π one π two, thatβs three times five, minus π two π one, thatβs negative five multiplied by π.
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The first and second elements simplify to zero.
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And the third element of the cross product of our two vectors becomes 15 plus five π.
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And thatβs because we have minus negative five.
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So we get plus five.
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Now, if we go back to the question, we see that the cross product of π and π is given to us.
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Weβre told itβs 50π€.
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Well, another way to represent that is using these sort of triangular brackets.
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The vector π crossed with π is zero, zero, 50.
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And we see then that, for the vectors to be equal, 50 must be equal to 15 plus five π.
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Weβll solve by subtracting 15 from both sides.
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And that gives us 35 equals five π.
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We then divide both sides of this equation by five.
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And we get seven equals π.
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And so if π is equal to three π’ minus five π£, π is equal to ππ’ plus five π£, and the cross product of π and π is 50π€, π must be equal to seven.