Given vector 𝐀 equals six, one, four and vector 𝐁 equals three, one, two, find 𝐀𝐁.
We recall that to find 𝐀𝐁, we subtract vector 𝐀 from vector 𝐁. The reason for this can be shown on a two-dimensional grid. This can then be transferred to three-dimensional vectors.
Let’s consider two points on the grid, A and B. The vector 𝐀 is given from its displacement from the origin O. The same is true of vector 𝐁. We wish to travel from point A to point B. This is the same as traveling from point A to point O and then point O to point B. Going from point A to point O would be equivalent to negative vector 𝐀. And going from point O to point B is equivalent to vector 𝐁. 𝐀𝐁 is equal to negative vector 𝐀 plus vector 𝐁. This can be rewritten as vector 𝐁 minus vector 𝐀.
We’re told in this question that vector 𝐀 is equal to six, one, four. This could also be written as a column vector as shown. It is also sometimes written in terms of vectors 𝐢, 𝐣, and 𝐤. In this case, six 𝐢 plus 𝐣 plus four 𝐤, where six, one, and four are the coefficients of 𝐢, 𝐣, and 𝐤.
For the purposes of this question, we will stick with the notation given. And we’re also told that vector 𝐁 is equal to three, one, two. In order to calculate 𝐁 minus 𝐀, we simply subtract the individual components. Three minus six is equal to negative three. One minus one is equal to zero. Finally, two minus four is equal to negative two. 𝐀𝐁 is therefore equal to negative three, zero, negative two.