Video Transcript
The points 𝐴 negative eight,
negative nine, negative two; 𝐵 zero, negative seven, six; and 𝐶 negative eight,
negative one, negative four form a triangle. Determine in vector form the
equation of the median drawn from 𝐶.
In this question, we have three
points 𝐴, 𝐵, and 𝐶, which are given in three-dimensional space. These three points we’re told form
a triangle. We’re told that there is a median
drawn from 𝐶, and so it would be useful to recall that a median is a line segment
joining a vertex to the midpoint of the opposite side. For example, if we drew this
two-dimensional triangle 𝐴𝐵𝐶, the median from 𝐶 would look like this.
Perhaps the best way to begin this
question is to see if we can find the point that is the midpoint of 𝐴𝐵. Let’s define this with the letter
𝑀. The formula to find the midpoint of
two points in space is very similar to that which we might use for two coordinates
in two-dimensional space. To find the midpoint 𝑀 of 𝑥 one,
𝑦 one, 𝑧 one and 𝑥 two, 𝑦 two, 𝑧 two, we have that 𝑀 is equal to 𝑥 one plus
𝑥 two over two, 𝑦 one plus 𝑦 two over two, 𝑧 one plus 𝑧 two over two. When we fill our values into this
formula, we need to make sure we’re using the values for 𝐴 and 𝐵 as, after all, we
need to find the midpoint of 𝐴𝐵.
Note that when we’re plugging in
our values, it doesn’t matter which point we use with our 𝑥 one, 𝑦 one, 𝑧 one
values or the 𝑥 two, 𝑦 two, 𝑧 two values. So we have that the midpoint 𝑀 is
equal to negative eight plus zero over two, negative nine plus negative seven over
two, and negative two plus six over two. Simplifying this, we have that 𝑀
is equal to negative four, negative eight, two. We can now clear some space so we
can begin to think about the vector form of the equation of this median. The vector form of an equation can
be written in the form 𝐫 equals 𝐫 sub zero plus 𝑡𝐯, where 𝐫 is the position
vector of a general point on the line, 𝐫 sub zero is the position vector of a given
point on the line, and 𝐯 is the direction vector. 𝑡 is a scalar multiple.
Let’s think about what would happen
if we model these three points in three-dimensional space. We’d have the triangle 𝐴𝐵𝐶 and
the median, which would be the line segment of 𝐶𝑀. So when it comes to writing the
median in vector form, the position vector can be the point 𝐶. But we still need to work out the
direction vector of 𝐂𝐌. To find the vector 𝐂𝐌, we
subtract the starting point 𝐶 from the terminal point 𝑀. So we have negative four subtract
negative eight, negative eight subtract negative one, and two subtract negative
four. Simplifying this, we have that
vector 𝐂𝐌 is equal to four, negative seven, six.
Now we have all the information
that we need to plug in to the vector form of the line. 𝐫 sub zero will be the position
vector representing point 𝐶. Vector 𝐯 will be represented by
the vector 𝐂𝐌. Therefore, the answer for the
equation of the median from 𝐶 is 𝐫 equals negative eight, negative one, negative
four plus 𝑡 four, negative seven, six.
