Video Transcript
The diagram shows a vector 𝐀 that has a vertical component with a magnitude of 130. The angle between the vector and the 𝑥-axis is 64 degrees. What is the magnitude of the vector? Give your answer to the nearest integer.
If we look at the diagram, we can see the vector 𝐀; the angle 64 degrees to the 𝑥-axis, which is the horizontal direction; and then the vertical component with the magnitude of 130. We’re looking for the magnitude of a vector, which means finding the magnitude along this direction. We’re going to solve this using trigonometry, so let’s first copy the vertical component over to the other side.
We know this vertical component has a magnitude of 130. And it now forms a right-angled triangle where we know the angle of 64 degrees, the opposite side, which has a magnitude of 130. And the value we want to find is the magnitude of the hypotenuse. Whenever we’re working with trigonometry, we need to recall the helpful mnemonic SOHCAHTOA, which tells us that if we have the opposite and we need the hypotenuse, the value we need is the sine of the angle. What this tells us is that the sin of an angle 𝜃 is equal to the opposite divided by the hypotenuse.
So if we enter our values into this, we have that the sin of 64 degrees is equal to the opposite, which is 130, divided by the hypotenuse. Now, let’s rearrange this in terms of the hypotenuse by multiplying both sides by the value of the hypotenuse, which will move this up here, and then dividing both sides by the sin of 64 degrees, which will move sin of 64 down here. And that gives us the hypotenuse equal to 130 divided by the sin of 64 degrees.
Now, if we put this into our calculators, making sure that they are in degrees, we get 144.638. And we’re asked for that value to the nearest integer, so that becomes 145. So the magnitude of the vector 𝐀 is equal to 145.
