Video Transcript
Determine 𝑚 given that the scalar product of the two vectors 𝚨 which is equal to 𝑚𝐢 minus six 𝐣 minus six 𝐤 and vector 𝚩 which is equal to negative two 𝐢 plus eight 𝐣 minus four 𝐤 is 27.
We recall that we can calculate the scalar or dot product of two vectors by multiplying their corresponding components and then finding the sum of these values. In this question, we are told that the scalar product is equal to 27. The 𝐢-components of vectors 𝚨 and 𝚩 are 𝑚 and negative two respectively, the 𝐣-components are negative six and eight, and the 𝐤-components are negative six and negative four. The scalar product of vector 𝚨 and vector 𝚩 is therefore equal to 𝑚 multiplied by negative two plus negative six multiplied by eight plus negative six multiplied by negative four. Multiplying 𝑚 and negative two give us negative two 𝑚. Negative six multiplied by eight is negative 48. And negative six multiplied by negative four is 24.
Our expression simplifies to negative two 𝑚 plus negative 48 plus 24. And we know this is equal to 27. The right-hand side simplifies to negative two 𝑚 minus 24. We can then add 24 to both sides of our equation. Negative two 𝑚 is therefore equal to 51. Our final step is to divide both sides of this equation by negative two. 𝑚 is therefore equal to negative 51 over two. As a half of 51 is 25.5, 𝑚 is equal to negative 25.5. This is the value of 𝑚 given that the scalar product of vectors 𝚨 and 𝚩 is 27.
