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If vector 𝐀 is equal to 𝑎𝐢 plus 𝐣 minus 𝐤 and the magnitude of vector 𝐀 is equal to the square root of six, find all the possible values of 𝑎.
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Before starting this question, it is worth noting that the wording says find all possible values of 𝑎.
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This suggests there will be more than one correct answer.
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We are given two pieces of information.
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We are told vector 𝐀 is equal to 𝑎𝐢 plus 𝐣 minus 𝐤 and the magnitude of vector 𝐀 is the square root of six.
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We know that for any vector written in the form 𝑥𝐢 plus 𝑦𝐣 plus 𝑧𝐤, then its magnitude is equal to the square root of 𝑥 squared plus 𝑦 squared plus 𝑧 squared.
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In this question, the square root of six is equal to the square root of 𝑎 squared plus one squared plus negative one squared.
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This is because the 𝐢-, 𝐣-, and 𝐤-components are 𝑎, one, and negative one, respectively.
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We can begin to solve this equation by squaring both sides.
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As squaring is the inverse or opposite of square rooting, the square root of six squared is equal to six.
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In the same way, the right-hand side becomes 𝑎 squared plus one squared plus negative one squared.
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Both one squared and negative one squared are equal to one.
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Therefore, this simplifies to six is equal to 𝑎 squared minus two.
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We can then subtract two from both sides of this equation so that 𝑎 squared is equal to four.
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Our final step is to square root both sides.
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The square root of 𝑎 squared is 𝑎.
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The square root of four is equal to two.
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But we must take the positive or negative of this.
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Therefore, 𝑎 is equal to positive or negative two.
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The possible values of 𝑎 such that the magnitude of vector 𝐀 is the square root of six are two and negative two.
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This is because when we square both of these, we get an answer of four.