𝐴 and 𝐵 are two points in an
orthonormal coordinate plane with a positive horizontal 𝑥-axis pointing to the
right and a positive vertical 𝑦-axis pointing up. The unit lengths of the axes are
given by the grid. If the coordinates of 𝐴 are one,
two, what are the coordinates of 𝐵?
Our points 𝐴 and 𝐵 are in an
orthonormal coordinate plane. And that’s a plane whose unit
lengths are given by the grid. Now remember, an orthonormal
coordinate plane is plane 𝑂; 𝐼, 𝐽 whose axes 𝑂𝐼 and 𝑂𝐽 are perpendicular,
that’s the 𝑥- and 𝑦-axes, and whose unit lengths 𝑂𝐼 and 𝑂𝐽 are equal. To find the coordinates of the
point 𝐵, we must first determine the position of the origin of the coordinate
plane. We can do this using the given
point 𝐴. We know that 𝐴 has coordinates
one, two. And this means that 𝐴 sits one
unit to the right of the origin and two units up from the origin. And so working backwards, the
origin is one unit left from 𝐴 and two units down from 𝐴.
So now if we find the position of
𝐵 with respect to this origin, we see that the point 𝐵 is one unit length left
from the origin 𝑂, that is, negative one unit, and is on the horizontal or 𝑥-axis,
which means we have zero units in the 𝑦-direction. The coordinates of 𝐵 are therefore
negative one, zero.