# Question Video: Finding the Components of a Vector from Its Magnitude and Argument Physics

The diagram shows a vector, π, that has a magnitude of 24. The angle between the vector and the π₯-axis is 43Β°. Give this vector in component form. Round all numbers in your answer to the nearest whole number.

03:35

### Video Transcript

The diagram shows a vector π that has a magnitude of 24. The angle between the vector and the π₯-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 π, which has a magnitude of 24 and makes an angle of 43 degrees with the π₯- or horizontal axis. Weβre asked to write π in component form, which means expressing it in the form π is equal to π sub π₯π’ hat plus π sub π¦π£ hat. π’ hat and π£ hat are unit vectors, where π’ hat represents one unit in the horizontal direction and π£ hat is one unit in the vertical direction.

The values that we need to find are π sub π₯, which is the magnitude or size of the horizontal component, and π sub π¦, 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 weβre given is 43 degrees. And relative to that angle, the side π sub π₯ is the adjacent side, π sub π¦ is the opposite side, and then the magnitude or size of the vector, which weβll call π΄, is the hypotenuse. Now weβre going to use trigonometry to solve this, so we need to recall SOHCAHTOA.

Letβs start by finding π sub π₯, 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 π equals adjacent divided by hypotenuse. So this gives us cos π is equal to π sub π₯ divided by π΄. And then we can multiply both sides by π΄. And that gives us π΄ cos π is equal to π sub π₯. Now we can put our numbers in. We have π΄ is equal to 24 and π is 43 degrees, which gives us 24 times the cos of 43 degrees. Now, if we put this into our calculator making sure itβs in degrees, we find that π sub π₯ is equal to 17.55. And weβre asked to give our answer to the nearest whole number, so that becomes 18. So π sub π₯ is equal to 18

Now letβs clear some space so we can work out π sub π¦. π sub π¦ is the opposite side of the triangle. And again we know the hypotenuse. So this time weβre going to work with the sine of the angle. So from this we recall that sin of π is equal to the opposite divided by the hypotenuse. Therefore, sin of π is equal to π sub π¦ divided by π΄. Now we can multiply both sides by π΄. And we have π΄ sin π is equal to π sub π¦. So substituting in our numbers we have π sub π¦ is equal to 24 times the sin of 43 degrees. Therefore, π sub π¦ is equal to 16.37. Again, we want this to the nearest whole number. So that becomes 16.

Now that we have both π sub π₯ and π sub π¦, we can write the vector π in component form as π is equal to 18π’ hat plus 16π£ hat. And that gives us our final answer.