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If 𝐴𝐵𝐶 is a triangle of area 248.5 square centimeters, find the value of the magnitude of the cross product of vectors 𝚩𝚨 and 𝚨𝐂.
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We will begin by sketching triangle 𝐴𝐵𝐶.
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We are told that it has an area of 248.5 square centimeters.
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And we recall that we can calculate the area of any triangle using the formula a half 𝑏𝑐 multiplied by sin 𝐴.
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In this question, we are asked to find the value of the magnitude of the cross product of vectors 𝚩𝚨 and 𝚨𝐂.
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We know that the magnitude of the cross product of two vectors 𝚨 and 𝚩 is equal to the magnitude of vector 𝚨 multiplied by the magnitude of vector 𝚩 multiplied by the magnitude of sin 𝜃.
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In this question, the magnitude of the cross product of vectors 𝚩𝚨 and 𝚨𝐂 is the magnitude of vector 𝚩𝚨 multiplied by the magnitude of 𝚨𝐂 multiplied by the magnitude of sin 𝜃, where 𝜃 is the angle between the two vectors.
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Extending the line 𝚩𝚨, we see that the angle 𝜃 between vectors 𝚩𝚨 and 𝚨𝐂 is as shown.
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This means that the measure of angle 𝜃 and the measure of angle 𝐴 sum to 180 degrees.
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And the measure of angle 𝜃 is therefore equal to 180 degrees minus the measure of angle 𝐴.
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The magnitude of vector 𝚩𝚨 is equal to the length of the side 𝐵𝐴 in our triangle.
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And we have labeled this lowercase 𝑐 as it is opposite angle 𝐶 in the triangle.
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Likewise, the magnitude of vector 𝚨𝐂 is equal to the length of side 𝐴𝐶, and this is equal to lowercase 𝑏.
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The cross product is therefore equal to 𝑐 multiplied by 𝑏 multiplied by the magnitude of sin of 180 degrees minus 𝐴.
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Since the sin of 180 degrees minus 𝜃 is equal to sin 𝜃, our expression is equal to 𝑐 multiplied by 𝑏 multiplied by sin 𝐴, which can be rewritten as 𝑏𝑐 multiplied by sin 𝐴.
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This is very similar to the formula for the area of a triangle.
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Since the area of our triangle is 248.5 square centimeters, then 248.5 is equal to a half 𝑏𝑐 multiplied by sin 𝐴.
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Multiplying both sides of this equation by two, we have 497 is equal to 𝑏𝑐 multiplied by sin 𝐴.
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We now have the exact same expression as in the cross product.
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And we can therefore conclude that the magnitude of the cross product of vectors 𝚩𝚨 and 𝚨𝐂 is 497.
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This leads us to a general formula for the area of a triangle.
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One-half of the magnitude of the cross product of vectors 𝚨 and 𝚩 is the area of a triangle that is spanned by vectors 𝚨 and 𝚩.