Question Video: Finding the Vertical Component of a Vector Physics • First Year of Secondary School

Video Transcript

The diagram shows a vector 𝚨 that has a magnitude of 43. The angle between the vector and the 𝑥-axis is 66 degrees. Work out the vertical component of the vector. Give your answer to the nearest integer.

We can see that in this question we’ve been given a diagram that shows a vector, that’s this black arrow here, and it’s labeled as 𝚨. This vector 𝚨 has a magnitude of 43 and the magnitude of a vector is the length of the arrow that represents it. So 43 is the length of this black arrow in the diagram. We’re told that the vector makes an angle of 66 degrees to the 𝑥-axis. We can see this in our diagram that the vector 𝚨 is at an angle of 66 degrees to the horizontal axis. This horizontal axis is our 𝑥-axis, while the vertical axis is our 𝑦-axis.

Now we’re being asked to find the vertical component of the vector 𝚨. The vertical or 𝑦-component of a vector tells us how far the vector extends in the 𝑦-direction. That is, if we trace across horizontally from the tip of the vector until we get to the 𝑦-axis, and that’s exactly what this black dashed line here is, then the 𝑦-component of the vector is the distance from the origin to the point where our dashed line meets the axis.

If the line we draw meets the 𝑦-axis in the positive 𝑦-direction relative to the origin like we’ve got here, then the 𝑦-component has a positive value. If instead our line met the axis somewhere in the negative 𝑦-direction from the origin, then the 𝑦-component of the vector would have a negative value. We can see that’s not the case here and that this line is in the positive 𝑦-direction relative to the origin. So we’ve got a positive vertical or 𝑦-component.

The magnitude of this vertical component is the distance between the origin and the point on the 𝑦-axis where our line meets it. We can notice that this length forms the vertical side of a right-angled triangle, and we’ll label it as 𝑎 subscript 𝑦. We know that this angle here between the vector 𝚨 and the 𝑥-axis is equal to 66 degrees. We can also recall that the 𝑥- and 𝑦-axes are perpendicular to each other, which means that they have an angle of 90 degrees between them.

So if we label this angle here as 𝜃, that’s the angle between the vector 𝚨 and the vertical or 𝑦-axis, then the angle 𝜃 and the angle of 66 degrees must add up to 90 degrees. If we subtract 66 degrees from both sides of this, we get an equation which says 𝜃 is equal to 90 degrees minus 66 degrees. And that works out as 24 degrees.

So in our right-angled triangle, we now know the value of this angle 𝜃. And we also know the length of the hypotenuse because that’s the magnitude 43 of our vector 𝚨. We can use this information to work out the length of the vertical side of the triangle, and we know that this is the 𝑦-component, or vertical component, of our vector.

In order to do this, we need to recall a trigonometric equation. Let’s consider a general right-angled triangle that has a hypotenuse of length ℎ, one angle of 𝜃, a side adjacent to this angle 𝜃 with a length 𝑎, and a side opposite the angle 𝜃 with a length 𝑜. For this general triangle, the cos of the angle 𝜃 is equal to 𝑎, the length of the side adjacent to 𝜃, divided by ℎ, the length of the hypotenuse.

Comparing the general right-angled triangle against the one that we’ve identified in our diagram, we can see that we know the values of both ℎ and 𝜃. ℎ, the length of the hypotenuse, is equal to 43, while we found that the angle 𝜃 is equal to 24 degrees. Meanwhile, 𝑎 subscript 𝑦 is our value for 𝑎, the adjacent side of the triangle.

Now in order to make use of this equation, we’re going to need to rearrange it to make 𝑎 the subject. To do this, we multiply both sides by ℎ. On the right-hand side, the ℎ in the numerator cancels the ℎ in the denominator. Then writing the equation the other way round, we have that 𝑎 is equal to ℎ times cos 𝜃. In our case, this adjacent side 𝑎 is equal to the vertical component of our vector. That’s 𝑎 subscript 𝑦. And then we can substitute our values for ℎ and 𝜃 into the right-hand side of this equation to give us that 𝑎 subscript 𝑦 is equal to 43 times the cos of 24 degrees.

Evaluating the expression gives a result of 39.282, where the ellipses are used to show that this has further decimal places. We’re told to give our answer to the nearest integer. By rounding our result, we get our answer to the question that the vertical component of the vector 𝚨 is equal to 39.