Find all the possible values of 𝑚 given vector 𝐀 is equal to negative four, three, one; vector 𝐁 is equal to six, negative six, 𝑚 minus 13; and the magnitude of vector 𝐀 plus vector 𝐁 is equal to seven.
We will begin this question by adding vector 𝐀 and vector 𝐁. We do this by adding the corresponding components. Negative four plus six is equal to two. Three plus negative six is equal to negative three. And one plus 𝑚 minus 13 is equal to 𝑚 minus 12. 𝐀 plus 𝐁 is equal to two, negative three, 𝑚 minus 12.
We are told that the magnitude of this vector is equal to seven. We can calculate the magnitude of any vector by finding the sum of the squares of the individual components and then square rooting our answer. The magnitude of 𝐀 plus 𝐁 is the square root of two squared plus negative three squared plus 𝑚 minus 12 squared. And we are told this is equal to seven.
Two squared is equal to four. Negative three squared is equal to nine. If we then square both sides of our equation, we have four plus nine plus 𝑚 minus 12 all squared is equal to 49. Subtracting four and nine from both sides of this equation gives us 𝑚 minus 12 all squared is equal to 36. We can then square root both sides of this equation such that 𝑚 minus 12 is equal to positive or negative the square root of 36.
The square root of 36 is equal to six. Therefore, 𝑚 minus 12 is equal to positive or negative six. This gives us two possible solutions: Either 𝑚 minus 12 is equal to six or 𝑚 minus 12 is equal to negative six. Adding 12 to both sides of both of these equations gives us 𝑚 is equal to 18 and 𝑚 is equal to six.
The two possible values of 𝑚 are therefore 18 and six.