Consider the vector 𝐕 with modulus three and an angle of 45 degrees measured counterclockwise from the positive 𝑥-axis. Using trigonometry, calculate the 𝑥- and 𝑦-components of the vector and, hence, write vector 𝐕 in the form 𝑥, 𝑦. Round your answer to two decimal places.
Looking at our diagram, we see that vector 𝐕 starts at the origin and ends at the blue dot. We want to write vector 𝐕 in the form 𝑥, 𝑦, in other words, solve for the 𝑥- and 𝑦-coordinates of this dot. We are told in the question that the modulus or length of the vector is equal to three and that the angle that the vector makes with the positive 𝑥-axis in the counterclockwise direction is 45 degrees. In order to calculate the 𝑥- and 𝑦-components of the vector, we are told to use trigonometry. And we can begin by creating a right triangle as shown. The 𝑥-component will be the horizontal distance along the 𝑥-axis as shown. And the 𝑦-component will be the vertical height of the triangle. The hypotenuse of the triangle will have a length of three, which is equal to the modulus of vector 𝐕.
We can now use our knowledge of the trigonometric ratios in right triangles to calculate the values of 𝑥 and 𝑦. And these can be recalled using the acronym SOH CAH TOA. The side labeled 𝑦 is opposite our given angle of 45 degrees. And the side labeled 𝑥 is adjacent or next to the 45-degree angle and the right angle. The sine ratio tells us that sin of angle 𝜃 is equal to the opposite over the hypotenuse. Substituting in our values, we have the sin of 45 degrees is equal to 𝑦 over three. 45 degrees is one of our special angles, and the sin of 45 degrees is root two over two. Multiplying through by three, we see that 𝑦 is equal to three root two over two.
We are asked to give our answer to two decimal places. So typing this into our calculator gives us 𝑦 is equal to 2.1213 and so on. To two decimal places, this is equal to 2.12.
The cosine ratio tells us that the cos of angle 𝜃 is equal to the adjacent over the hypotenuse. This time, we have the cos of 45 degrees is equal to 𝑥 over three. And as the cos of 45 degrees is also equal to root two over two, 𝑥 is equal to three root two over two.
Both the 𝑥- and 𝑦-components are therefore equal to 2.12 to two decimal places. And we can therefore write the vector 𝐕 in the Cartesian form 𝑥, 𝑦 as 2.12, 2.12.