Video Transcript
Given that 𝐴 is the point one, zero, two; 𝐵 is the point two, two, two; and 𝐶 is the point two, one, two, find the vector from 𝐀 to 𝐁 added to the vector from 𝐀 to 𝐂.
In this question, we’re given the coordinates of three different points in three dimensions, the points 𝐴, 𝐵, and 𝐶. We need to use this to determine the vector from 𝐀 to 𝐁 added to the vector from 𝐀 to 𝐂. We’ll start by recalling how we find the vector between two points. The vector from one point to another represents the displacement when moving from the first point to the second point. We can find this by subtracting the position vector of vector 𝐀 from the position vector of vector 𝐁 because this would then tell us the change in the 𝑥-coordinate, the change in the 𝑦-coordinate, and the change in the 𝑧-coordinate when moving from point 𝐴 to point 𝐵.
In our case, that’s going to be the vector two, two, two minus the vector one, zero, two. Remember, to subtract two vectors, all we need to do is subtract the corresponding components of each vector. This gives us the vector two minus one, two minus zero, two minus two. Evaluating the expressions in each of the components of this vector gives us the vector one, two, zero. What this really tells us is the 𝑥-coordinate of point 𝐵 is one higher than the 𝑥-coordinate of point 𝐴. The 𝑦-coordinate of point 𝐵 is two higher than the 𝑦-coordinate of point 𝐴. And points 𝐴 and 𝐵 have the same 𝑧-coordinate.
We can then do exactly the same to find the vector from 𝐀 to 𝐂. Once again, we want to find the difference in the position vectors of point 𝐶 and point 𝐴. That’s the vector two, one, two minus the vector one, zero, two. And we do this by subtracting the vectors component-wise. This gives us the vector one, one, zero. Now that we found the vector from 𝐀 to 𝐁 and the vector from 𝐀 to 𝐂, we’re ready to add these two vectors together. This means we need to add the vector one, two, zero to the vector one, one, zero.
Remember, when we add two vectors of the same dimensions together, we just add the corresponding components together. Doing this, we get the vector one plus one, two plus one, zero plus zero. And if we evaluate the expressions for each of our components, we get the vector two, three, zero which is our final answer. Therefore, given the point 𝐴 one, zero, two; the point 𝐵 two, two, two; and the point 𝐶 two, one, two, we were able to show the vector from 𝐀 to 𝐁 plus the vector from 𝐀 to 𝐂 is the vector two, three, zero.
