Video Transcript
Consider the vector negative two,
three. Calculate the direction of the
vector, giving your solution as an angle to the nearest degree measured
counterclockwise from the positive 𝑥-axis.
Since we’re given the components of
our vector, let’s start out by plotting it on this 𝑥𝑦-plane. We’ll say that each of these tick
marks represents a distance of one unit. And we’re told that our vector has
an 𝑥-component of negative two and a 𝑦-component of positive three. That tells us that with its tail at
the origin our vector will look like this. We want to calculate the direction
of this vector, and we’ll do it in terms of an angle that begins at the positive
𝑥-axis and goes to our vector. We can give this angle a name. We’ll call it 𝜃. At this point, let’s recall that in
general when a vector, we can call it 𝐕, is given in polar form, then that vector
is defined by a radial distance from the origin and an angle from the positive
𝑥-axis.
In our exercise, we want to solve
for this angle 𝜃. And to do this, we’re given a
vector not in polar form but in rectangular form. In using this 𝑥 and 𝑦 information
then to solve for 𝜃, we’re effectively converting from a rectangular form of a
vector to part of its polar form. We say all that because the tan of
𝜃, where 𝜃 is the angle of a vector in its polar form, is equal to the ratio of
the rectangular components of that vector, 𝑦 to 𝑥. This tells us that the tan of our
angle of interest, 𝜃, is equal to three divided by negative two. And then if we take the inverse or
arctan of both sides, the left-hand side simplifies to the angle we want to solve
for.
Now, if we go and evaluate this
expression on our calculator to the nearest hundredth of a degree, we find a result
of negative 56.31 degrees. Looking at our sketch though, we
see that this can’t be the angle 𝜃. What’s happening here is because
the 𝑥-value in this fraction is negative — in other words, the vector we’re working
with occupies either the second or the third quadrant — to correctly compute our
angle, we’ll need to take our calculated result and add 180 degrees to it. For any vector with a negative
𝑥-component like we have here, this is a standard practice for accurately
calculating its direction.
And so now if we add these two
angles together, we get a result of 123.69 degrees. And recalling that we give our
answer rounded to the nearest degree, we can see that 𝜃 equals 124 degrees. This is the direction of our vector
to the nearest degree measured counterclockwise from the positive 𝑥-axis.
