Video Transcript
What is the distance between the
point 19, five, five and the 𝑥-axis?
Any point that lies on the 𝑥-axis
will have coordinates 𝑥, zero, zero. Both the 𝑦- and 𝑧-coordinates
must be equal to zero. We’re given the coordinates of a
point 19, five, five. The point on the 𝑥-axis that is
closest to this will have coordinates 19, zero, zero. The shortest distance will be to
the point where the 𝑥-coordinate is the same. We know that we can calculate the
distance between two points in three dimensions using an adaption of the Pythagorean
theorem. If we have two points with
coordinates 𝑥 one, 𝑦 one, 𝑧 one and 𝑥 two, 𝑦 two, 𝑧 two, the distance between
them is equal to the square root of 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦
one squared plus 𝑧 two minus 𝑧 one squared.
Substituting in our two coordinates
gives us the square root of 19 minus 19 squared plus zero minus five squared plus
zero minus five squared. 19 minus 19 is equal to zero. Zero minus five is equal to
negative five. So we are left with the square root
of negative five squared plus negative five squared. Multiplying a negative number by a
negative number gives us a positive answer. Therefore, negative five squared is
equal to 25. This means that our answer
simplifies to the square root of 50.
It is worth pointing out that we
could have subtracted the coordinates in the other order as five minus zero squared
is also equal to 25. As squaring a number always gives a
positive answer, it doesn’t matter which order we subtract our coordinates in. We can actually simplify our answer
by using our laws of radicals or surds. The square root of 50 is equal to
the square root of 25 multiplied by the square root of two. As the square root of 25 equals
five, we’re left with five multiplied by the square root of two or five root
two. The square root of 50 is equal to
five root two. We can therefore conclude that the
distance between the points 19, five, five and the 𝑥-axis is five root two length
units.
We might actually notice a shortcut
here. To find the distance between any
point and an axis, we simply find the sum of the squares of the other two
coordinates and then square root the answer. As we want to calculate the
distance to the 𝑥-axis, we square the 𝑦- and 𝑧-coordinates, find their sum, and
then square root our answer. If we needed to calculate the
distance between a point and the 𝑦-axis, we would square the 𝑥- and
𝑧-coordinates, find the sum of these, and then square root that answer. We would use the same method to
find the distance between a point and the 𝑧-axis, this time using the 𝑥- and
𝑦-coordinates.
