Video: Finding the Value of an Unknown Component of a Vector Using its Magnitude

If 𝐀 = 𝑎𝐢 + 𝐣 − 𝐤 and |𝐀| = √6, find all the possible values of 𝑎.

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Video Transcript

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.

Before starting this question, it is worth noting that the wording says find all possible values of 𝑎. This suggests there will be more than one correct answer. We are given two pieces of information. 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. 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. In this question, the square root of six is equal to the square root of 𝑎 squared plus one squared plus negative one squared. This is because the 𝐢-, 𝐣-, and 𝐤-components are 𝑎, one, and negative one, respectively.

We can begin to solve this equation by squaring both sides. As squaring is the inverse or opposite of square rooting, the square root of six squared is equal to six. In the same way, the right-hand side becomes 𝑎 squared plus one squared plus negative one squared. Both one squared and negative one squared are equal to one. Therefore, this simplifies to six is equal to 𝑎 squared minus two. We can then subtract two from both sides of this equation so that 𝑎 squared is equal to four.

Our final step is to square root both sides. The square root of 𝑎 squared is 𝑎. The square root of four is equal to two. But we must take the positive or negative of this. Therefore, 𝑎 is equal to positive or negative two. The possible values of 𝑎 such that the magnitude of vector 𝐀 is the square root of six are two and negative two. This is because when we square both of these, we get an answer of four.

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