Video Transcript
The diagram shows a vector 𝐀
that has a magnitude of 22. The angle between the vector
and the 𝑥-axis is 36 degrees. Work out the horizontal
component of the vector. Give your answer to two
significant figures.
Alright, we see this vector 𝐀
sketched out. And we’re told it has a length
or magnitude of 22. Along with this, we know that
the vector forms an angle of 36 degrees with the positive 𝑥-axis. Our goal is to solve for its
horizontal component. This is equal to the horizontal
projection of our vector onto this axis.
As we go about solving for the
length of this orange line, let’s note that our dashed line intersects our
horizontal axis at a right angle. In other words, we have here a
right triangle. Here’s the hypotenuse, here’s
another side, and here’s the third. In solving then for the
horizontal component of our vector, we’re solving for one of the sides of this
right triangle. We can do this using
trigonometry.
Let’s remember that, given a
right triangle, if we know one of the other interior angles, then we can define
the sides of this right triangle as hypotenuse ℎ, the side opposite our angle 𝜃
𝑜, and the side adjacent to that angle 𝑎. Set up this way, it’s the
adjacent side, what we’ve called 𝑎 over here, that we want to solve for to
figure out our horizontal component.
Now, if we were to take the cos
of this angle 𝜃, then that would equal the ratio of our adjacent side length to
our hypotenuse. Or multiplying both sides of
this equation by the hypotenuse, canceling that factor out on the right, we have
that the adjacent side of our right triangle equals the cos of 𝜃 times ℎ. This relates to our situation
with vector 𝐀 because in this case we know the length of our hypotenuse and we
also know this angle. So we can actually say that the
length of our triangle’s hypotenuse, 22, multiplied by the cos of our angle of
36 degrees is equal to what we’ll call 𝐴 sub 𝑥, the horizontal component of
the vector 𝐀. When we enter this expression
on our calculator and keep two significant figures, our answer is 18. This is the horizontal
component of the vector 𝐀.
