Video Transcript
Consider the points 𝐴, 𝐵, and 𝐶
in the figure. Choose the parametric equations of
the line from 𝐴 to 𝐵. Option (A) 𝑥 is equal to 𝑘 and 𝑦
is equal to one. Option (B) 𝑥 is equal to zero and
𝑦 is equal to 𝑘. Option (C) 𝑥 is equal to 𝑘 and 𝑦
is equal to zero. Option (D) 𝑥 is equal to one and
𝑦 is equal to two. Or option (E) 𝑥 is equal to one,
𝑦 is equal to one plus 𝑘. Choose the parametric equations of
the line from 𝐵 to 𝐶. Option (A) 𝑥 is equal to 𝑘 minus
three and 𝑦 is equal to two. Option (B) 𝑥 is equal to 𝑘 plus
one and 𝑦 is equal to three. Option (C) 𝑥 is equal to two and
𝑦 is equal to 𝑘 minus three. Option (D) 𝑥 is equal to three and
𝑦 is equal to 𝑘 minus two. Or option (E) 𝑥 is equal to two
and 𝑦 is equal to three. Choose the parametric equations of
the line from 𝐴 to 𝐶. Option (A) 𝑥 is equal to two 𝑘
plus one and 𝑦 is equal to two 𝑘 plus one. Option (B) 𝑥 is equal to 𝑘 minus
three and 𝑦 is equal to two. Option (C) 𝑥 is equal to two 𝑘
minus one and 𝑦 is equal to three 𝑘 minus one. Option (D) 𝑥 is equal to two and
𝑦 is equal to 𝑘 minus two. Or option (E) 𝑥 is equal to two
and 𝑦 is equal to two.
In this question, we’re given a
coordinate grid and three points on this grid, 𝐴, 𝐵, and 𝐶. We need to use this figure to
determine parametric equations for the three lines between the pairs of points given
in the figure.
And there’s a few different ways of
doing this. Let’s start by recalling what we
mean by the parametric equations of a line. They’re two equations of the form
𝑥 is equal to 𝑎 times 𝑘 plus 𝑥 sub zero and 𝑦 is equal to 𝑏 times 𝑘 plus 𝑦
sub zero. And we can recall we can choose any
point 𝑥 sub zero, 𝑦 sub zero which lies on the line and any nonzero vector 𝑎, 𝑏
which is parallel to the line. These will give us equivalent
parametric equations for the line. For example, in the line between 𝐴
and 𝐵, we know that both points 𝐴 and 𝐵 lie on the line. So, we can choose either of these
points to give us our values of 𝑥 sub zero or 𝑦 sub zero, or we could choose any
point on this line.
So, instead of choosing this point,
let’s instead find a nonzero vector parallel to the line between 𝐴 and 𝐵. And we can do this in two different
ways. Either we can notice that the
vector from 𝐴 to 𝐵 is parallel to our line and we can calculate the vector from 𝐴
to 𝐵 by subtracting the position vector of 𝐀 from the position vector of 𝐁 or,
alternatively, we can directly find the components of this vector from the
diagram. We see that this line is vertical,
so it has no horizontal component, and we can see we move two units up. In either case, we show that the
vector from 𝐴 to 𝐵 is the vector zero, two. However, remember we can choose any
nonzero vector parallel to the line for our vector 𝐀𝐁.
So, instead of choosing the vector
zero, two, let’s instead choose the vector zero, one. Our value of 𝐀 will be zero, and
our value of 𝐁 will be one. And this is exactly the same as
noting our line is vertical, so it’s parallel to the unit directional vector 𝐣. If we substitute 𝐀 is equal to
zero and 𝐁 is equal to one into our parametric equations, we see that 𝑥 will be
equal to 𝑥 sub zero and 𝑦 will be equal to 𝑘 plus 𝑦 sub zero. So, we can eliminate any options
which don’t have 𝑥 equal to some constant value. These will not be vertical
lines. And remember here 𝑘 is a
scalar. It can take on many values. Options (A) and (C) cannot be
correct.
Next, remember our value of 𝑦 can
vary since this is a vertical line, so we can’t have a zero coefficient of 𝑘 in the
𝑦 equation. Our values of 𝑦 can’t be constant,
so option (D) cannot be correct. This leaves us with two remaining
options, and we can show that option (B) cannot be the correct answer. And to do this, let’s consider the
values of 𝑥 sub zero, 𝑦 sub zero in equation (B). The value of 𝑥 sub zero is zero,
and the value of 𝑦 sub zero is also zero. So, the point with coordinates
zero, zero must lie on the parametric equations given in option (B). However, we can see that our
vertical line does not pass through the origin, so this cannot be the correct
answer.
Instead, let’s choose the point 𝐴
which has coordinates one, one to be our point 𝑥 sub zero, 𝑦 sub zero. This means we substitute 𝐴 is
equal to zero, 𝐵 is equal to one, 𝑥 sub zero is equal to one, and 𝑦 sub zero is
equal to one into our parametric equations. We get 𝑥 is equal to one and 𝑦 is
equal to one plus 𝑘, which we can see is option (E). Let’s now clear some space and
follow a similar process to determine the parametric equations of the line passing
through points 𝐵 and 𝐶. This time, we could see on our
diagram that the line between 𝐵 and 𝐶 is a horizontal line.
Once again, we can choose either
point 𝐵 or point 𝐶 or any point on this line to be our point with coordinates 𝑥
sub zero, 𝑦 sub zero. And we can find a vector parallel
to the line in many different ways. We could find the vector from 𝐵 to
𝐶, or we can note this is a horizontal line, so it’s parallel to the unit
directional vector 𝐢. In other words, it’s parallel to
the vector one, zero.
So, we’ll choose our value of 𝐴
equal to one and our value of 𝐵 equal to zero. And since the value of 𝐵 is zero,
we can note in our parametric equations the value of 𝑦 must be a constant
value. Options (C) and (D) are
incorrect. We can also note that option (E) is
incorrect since this states that 𝑥 is a constant value. In fact, there’s only one solution
to these parametric equations. It’s the single point with
coordinates two, three. It’s not a horizontal line.
This leaves us with two possible
answers. And to determine if either of these
are possible equations of the line between 𝐵 and 𝐶, we need to determine the
coordinates of the point 𝑥 sub zero, 𝑦 sub zero. And one way of doing this is to
note the 𝑦-coordinate of every single point on our line is three. In equation (A), we can see that
our value of 𝑦 is constant. However, it’s a constant value of
two. These are actually the parametric
equations of the horizontal line 𝑦 is equal to two. However, we want the horizontal
line 𝑦 is equal to three. So, (A) is not the correct answer,
and we need 𝑦 to be the constant value of three, which is given in option (B). And although it’s not necessary to
conclude this is the correct answer, we can note that the point chosen for 𝑥 sub
zero, 𝑦 sub zero in these parametric equations is one, three — the point 𝐵.
Therefore, substituting 𝐴 is equal
to one, 𝐵 is equal to zero, 𝑥 sub zero is equal to one, and 𝑦 sub zero is equal
to three into the parametric equations, we can show that the parametric equations of
the line between 𝐵 and 𝐶 is 𝑥 is equal to 𝑘 plus one and 𝑦 is equal to three,
which is option (B).
Let’s now do this one final time
for the line between 𝐴 and 𝐶. Let’s start by finding a vector
parallel to this line. We can do this from the
diagram. However, in general, we’ll want to
use the fact that the vector from 𝐴 to 𝐂 is equal to the position vector of 𝐂
minus the position vector of 𝐀. And we can recall the components of
the position vector of a point are equal to its coordinates. So, the position vector of 𝐂 is
the vector three, three. And the position vector of 𝐀 is
the vector one, one, since 𝐂 has coordinates three, three and 𝐀 has coordinates
one, one.
And now, we evaluate the vector
subtraction. We do this componentwise. We get the vector three minus one,
three minus one, which we can evaluate is the vector two, two. This is the same as saying for
every two units we move across on our line, we move two units upwards. We can see this in the diagram.
We could use the same reasoning we
did before to rewrite this as the vector one, one for our parametric equations. However, if we look at the five
given options, we can see in none of these five options are both coefficients of 𝑘
in the equation is equal to one. In option (A), both of these
coefficients are two. In option (B), the coefficient of
𝑦 is zero. In option (C), the coefficients of
𝑘 are two and three. In option (D), the coefficient of
𝑘 in the 𝑥 equation is zero. And in option (E), the coefficients
of 𝑘 in both equations are zero.
So, the only equation where both
coefficients of 𝑘 are the same is option (A). And in fact, this is enough to
exclude all four of the other given options since remember the vector 𝐀𝐁 must be
parallel to the given line. And remember this also can’t be the
zero vector. So, the values of 𝐀and 𝐁 are the
same and they’re both nonzero. And this is only true for option
(A). And we can also, for due diligence,
check the value of the point 𝑥 sub zero, 𝑦 sub zero. It’s the point with coordinates
one, one, which we can see is point 𝐴.
Therefore, if we substitute 𝑎 is
equal to two, 𝑏 is equal to two, 𝑥 sub zero is equal to one, and 𝑦 sub zero is
equal to one into our parametric equations, we get 𝑥 is equal to two 𝑘 plus one
and 𝑦 is equal to two 𝑘 plus one, which we can see is option (A).
