The diagram shows a vector 𝐀 that has a magnitude of 24. The angle between the vector and the 𝑥-axis is 43 degrees. Give this vector in component form. Round all numbers in your answer to the nearest whole number.
In the diagram, we can see the vector 𝐀, which has a magnitude of 24 and makes an angle of 43 degrees with the 𝑥- or horizontal axis. We’re asked to write 𝐀 in component form, which means expressing it in the form 𝐀 is equal to 𝑎 sub 𝑥𝐢 hat plus 𝑎 sub 𝑦𝐣 hat. 𝐢 hat and 𝐣 hat are unit vectors, where 𝐢 hat represents one unit in the horizontal direction and 𝐣 hat is one unit in the vertical direction.
The values that we need to find are 𝑎 sub 𝑥, which is the magnitude or size of the horizontal component, and 𝑎 sub 𝑦, which is the magnitude or size of the vertical component. If we draw these onto our diagram, they make two sides of a right-angled triangle. The angle of the triangle that we’re given is 43 degrees. And relative to that angle, the side 𝑎 sub 𝑥 is the adjacent side, 𝑎 sub 𝑦 is the opposite side, and then the magnitude or size of the vector, which we’ll call 𝐴, is the hypotenuse. Now we’re going to use trigonometry to solve this, so we need to recall SOHCAHTOA.
Let’s start by finding 𝑎 sub 𝑥, which is the adjacent side. We already know the hypotenuse, so this tells us we need to use the cosine of the angle. So we need to recall that cos 𝜃 equals adjacent divided by hypotenuse. So this gives us cos 𝜃 is equal to 𝑎 sub 𝑥 divided by 𝐴. And then we can multiply both sides by 𝐴. And that gives us 𝐴 cos 𝜃 is equal to 𝑎 sub 𝑥. Now we can put our numbers in. We have 𝐴 is equal to 24 and 𝜃 is 43 degrees, which gives us 24 times the cos of 43 degrees. Now, if we put this into our calculator making sure it’s in degrees, we find that 𝑎 sub 𝑥 is equal to 17.55. And we’re asked to give our answer to the nearest whole number, so that becomes 18. So 𝑎 sub 𝑥 is equal to 18
Now let’s clear some space so we can work out 𝑎 sub 𝑦. 𝑎 sub 𝑦 is the opposite side of the triangle. And again we know the hypotenuse. So this time we’re going to work with the sine of the angle. So from this we recall that sin of 𝜃 is equal to the opposite divided by the hypotenuse. Therefore, sin of 𝜃 is equal to 𝑎 sub 𝑦 divided by 𝐴. Now we can multiply both sides by 𝐴. And we have 𝐴 sin 𝜃 is equal to 𝑎 sub 𝑦. So substituting in our numbers we have 𝑎 sub 𝑦 is equal to 24 times the sin of 43 degrees. Therefore, 𝑎 sub 𝑦 is equal to 16.37. Again, we want this to the nearest whole number. So that becomes 16.
Now that we have both 𝑎 sub 𝑥 and 𝑎 sub 𝑦, we can write the vector 𝐀 in component form as 𝑎 is equal to 18𝐢 hat plus 16𝐣 hat. And that gives us our final answer.