### Video Transcript

Given vector 𝐀 is equal to 𝑎,
negative 10, negative nine and 𝐁 is equal to negative three, 𝑏, three, if 𝐴𝐵
equals five, three, 𝑐, find the value of 𝑎 plus 𝑏 plus 𝑐.

We begin this question by recalling
that 𝐴𝐵 is equal to vector 𝐁 minus vector 𝐀. We could rewrite each of the
vectors in terms of 𝐢, 𝐣, and 𝐤. For example, vector 𝐀 is equal to
𝑎𝐢 minus 10𝐣 minus nine 𝐤. In this case, we will write them as
column vectors, as shown. We can then solve each of the rows
individually.

On the top row, we have the
equation five equals negative three minus 𝑎. Adding three to both sides gives us
eight is equal to negative 𝑎. Dividing both sides by negative one
gives us a value of 𝑎 equal to negative eight. The second row gives us the
equation three is equal to 𝑏 minus negative 10. Subtracting a negative number is
the same as adding the absolute value of that number. So, the equation simplifies to
three equals 𝑏 plus 10. We can then subtract 10 from both
sides, giving us a value of 𝑏 equal to negative seven. Finally, the bottom row of the
vectors gives us the equation 𝑐 is equal to three minus negative nine. This is the same as three plus
nine. So, 𝑐 is equal to 12.

We have now calculated values of
𝑎, 𝑏, and 𝑐, which we can use to find the value of 𝑎 plus 𝑏 plus 𝑐. We need to add negative eight,
negative seven, and 12. Negative eight plus negative seven
is equal to negative 15. Adding 12 to this gives us negative
three.

The value of 𝑎 plus 𝑏 plus 𝑐
equals negative three.