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.