If vector 𝐀 is equal to negative three 𝐢 plus three 𝐣 plus four 𝑚𝐤, vector 𝐁 is equal to negative four 𝐢 minus six 𝐣 minus seven 𝐤, and vectors 𝐀 and 𝐁 are perpendicular, find the value of 𝑚.
The notation in the question tells us that vectors 𝐀 and 𝐁 are perpendicular. We recall that if any two vectors 𝐮 and 𝐯 are perpendicular, then their dot or scalar product is equal to zero. We will use this fact to help calculate the value of 𝑚.
We can calculate the dot or scalar product of vectors 𝐀 and 𝐁 by multiplying their corresponding components and then finding the sum of these values. The 𝐢-components of vectors 𝐀 and 𝐁 are negative three and negative four, respectively. The 𝐣-components are three and negative six. The 𝐤-components are four 𝑚 and negative seven. The dot product of vectors 𝐀 and 𝐁 is therefore equal to negative three multiplied by negative four plus three multiplied by negative six plus four 𝑚 multiplied by negative seven.
Negative three multiplied by negative four is 12. Three multiplied by negative six is negative 18. And four 𝑚 multiplied by negative seven is negative 28𝑚. Our expression simplifies to 12 plus negative 18 plus negative 28𝑚. As the vectors are perpendicular, this is equal to zero.
12 plus negative 18 is negative six. Therefore, zero is equal to negative six minus 28𝑚. We can add 28𝑚 to both sides of this equation such that 28𝑚 is equal to negative six. Dividing through by 28, we see that 𝑚 is equal to negative six over 28. As both the numerator and denominator are divisible by two, this simplifies to negative three over 14 or negative three fourteenths. This is the value of 𝑚 such that the vectors 𝐀 and 𝐁 are perpendicular.