WEBVTT
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Find the direction cosines of the vector 𝐀 with components five, two, and eight.
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We recall that if vector 𝐯 has components 𝐯 sub 𝑥, 𝐯 sub 𝑦, and 𝐯 sub 𝑧 and direction angles 𝛼, 𝛽, and 𝛾, then the direction cosines cos of 𝛼, cos of 𝛽, and cos of 𝛾 are equal to 𝐯 sub 𝑥 over the magnitude of 𝐯, 𝐯 sub 𝑦 over the magnitude of 𝐯, and 𝐯 sub 𝑧 over the magnitude of 𝐯, respectively, where the magnitude of vector 𝐯 is equal to the square root of 𝐯 sub 𝑥 squared plus 𝐯 sub 𝑦 squared plus 𝐯 sub 𝑧 squared.
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In this question, we are told that vector 𝐀 has components five, two, and eight.
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The magnitude of vector 𝐀 is therefore equal to the square roots of five squared plus two squared plus eight squared.
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Five squared is equal to 25, two squared is equal to four, and eight squared is equal to 64.
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Summing these three values gives us an answer of 93.
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Therefore, the magnitude of vector 𝐀 is the square root of 93.
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This means that the cos of angle 𝛼 is equal to five over the square root of 93, the cos of angle 𝛽 is equal to two over the square root of 93, and, finally, the cos of angle 𝛾 is equal to eight over the square root of 93.
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The direction cosines of vector 𝐀 are five over the square root of 93, two over the square root of 93, and eight over the square root of 93.