Video Transcript
The lines 𝑥 plus 𝑦 plus four is
equal to zero and 𝐫 is equal to the vector negative one, four plus 𝑘 times the
vector two, two intersect orthogonally. Find the coordinates of the point
of intersection.
In this question, we’re given the
equation of two lines: one given in general form and one given in vector form. We’re given two pieces of
information about these two lines. We’re told that they intersect, and
we′re also told that this intersection is orthogonal. We need to use this information to
determine the coordinates of the point of intersection.
To do this, we can start by
recalling when we say that two lines intersect orthogonally, this means that they
intersect at right angles. In other words, the angle between
the two lines is 90 degrees. And we might be tempted to start
using this information to answer this question. For example, we know if two
straight lines were neither vertical or perpendicular, then the product of their
slopes will be negative one. However, this is not necessary to
answer this question, and in fact it won′t help us answer this question. We only need to use the fact that
these two lines intersect.
Instead, we can recall that to find
the point of intersection between two straight lines, we want to solve them as
simultaneous equations. Of course, we can′t do this yet
because our second line is given in vector form. So we′re going to start by
converting this into a different form. We can do this by first recalling
how we find the slope of a straight line given in vector form. If this line has direction vector
𝐝, which is the vector 𝑎, 𝑏, and the value of 𝑎 is nonzero, then the slope of
this line is given by 𝑏 divided by 𝑎, the change in 𝑦 divided by the change in
𝑥. In our case, we can see the
direction vector of our line is two, two. In other words, for every two units
we move to the right on this line, we move two units upwards. The slope 𝑚 is given by two
divided by two, which is equal to one.
At this point, let′s quickly think
about the equation of the other line. We could rearrange this equation to
be in slope–intercept form. That′s 𝑦 is equal to negative 𝑥
minus four, which tells us that the slope of this other line is negative one,
because that′s the coefficient of 𝑥 in this equation. Recall that multiplying the slopes
of two perpendicular lines will give us a result of negative one. And indeed we can note that
negative one multiplied by one is equal to negative one. So this checks out with the
information we were given in the question. These two lines are orthogonal.
We can also recall that we can find
the coordinates of a point which lies on the vector form of the equation of a line
by substituting the parameter equal to zero. So we substitute 𝑘 is equal to
zero into the vector form of the equation of the line. And we see that the point with
position vector negative one, four lies on this line, which of course tells us the
line passes through the point with coordinates negative one, four.
We now have the slope of this line
and the coordinates of a point which it passes through. So we can determine the equation of
the straight line by using the point–slope form. We recall this tells us the
equation of a line of slope 𝑚 passing through the point with coordinates 𝑥 sub
one, 𝑦 sub one is 𝑦 minus 𝑦 sub one is equal to 𝑚 times 𝑥 minus 𝑥 sub one. And for this line, our value of 𝑚
is one, 𝑥 sub one is negative one, and 𝑦 sub one is four. We can substitute these values into
the equation. We get 𝑦 minus four is equal to
one times 𝑥 minus negative one.
Now we can simplify this
equation. First, multiplying by one doesn′t
change the value, and subtracting negative one is the same as adding one. So we have 𝑦 minus four is equal
to 𝑥 plus one. Now, we add four to both sides of
the equation to get that 𝑦 is equal to 𝑥 plus five. So we′ve now found the equation of
our second line, and we already know the equation of the first line.
We now need to solve these as
simultaneous equations to determine the point of intersection. We need to solve the simultaneous
equations 𝑥 plus 𝑦 plus four is equal to zero and 𝑦 is equal to 𝑥 plus five. And there are many different ways
of solving simultaneous equations. So we’ll only go through one of
these. Since we already have an expression
for 𝑦, we′re going to substitute this expression into the first equation. This will allow us to eliminate the
𝑦-variable. Doing this, we get 𝑥 plus 𝑥 plus
five plus four must be equal to zero. Simplifying the left-hand side of
this equation, we get two 𝑥 plus nine is equal to zero.
We can now solve this equation for
𝑥. We subtract nine from both sides of
the equation to get two 𝑥 is equal to negative nine. And then we divide through by
two. 𝑥 is equal to negative nine
divided by two. This is the 𝑥-coordinate of the
point of intersection between the two curves. We can substitute this value into
either of our original two equations to determine the value of 𝑦. We’ll substitute this into the
second equation. We get that 𝑦 is equal to negative
nine over two plus five. And we can evaluate this. Five is equal to 10 divided by
two. So we get negative nine over two
plus 10 over two, which is one-half. And this gives us our final
answer.
Therefore, we were able to show the
coordinates of the points of intersection between the lines 𝑥 plus 𝑦 plus four is
equal to zero and 𝐫 is equal to the vector negative one, four plus 𝑘 times the
vector two, two is negative nine over two, one-half.
