The Geometric Representation of Vectors - Problem 2

Let's plot two more vectors and find their lengths, starting with vector QR. Vector QR starts at point Q,(-2, 1) and terminates at point R,(-2, 6). I’ve plotted those two points here on the coordinate system.

So starting at Q and ending at R looks like this. It’s a vertical vector terminating at R. So that’s where the arrow head goes. I need to find its length. Finding the length of a vertical or a horizontal vector is really easy. You don’t need the Pythagorean theorem, you just need to count. 1, 2, 3, 4, 5, units. The length of vector QR is 5.

Let’s do another one; vector OS. Where O is the point (0, 0), the origin and S is point (4, 4). This is vector OS terminating at S and to find its length I will use the Pythagorean Theorem. I’ll draw a horizontal leg that ends just beneath point S and then a vertical leg. Clearly the length of the horizontal leg is 4. The vertical leg also has length 4. In fact that’s pretty clear because I’m staring at point (0, 0) and this point is (4, 4) you know that these legs are 4 and 4.

Let’s use the Pythagorean Theorem to find the length. Square root of 4² plus 4². That’s the square root of 16 plus 16, or 32. The largest perfect square factor of 32 is 16, whose square root is 4. So this becomes 4 root 2 and that’s our answer. The length of vector OS is 4 root 2.