If the magnitude of vector 𝐀 is equal to six centimeters, then vector 𝐀 is equal to what. Is it (A) root three, three; (B) three, three root three; (C) three root three, three; or (D) six, six root three?
In this question, we want to find the rectangular form of vector 𝐀 using a graphical representation and a given length or magnitude of a vector. We recall that any vector 𝐕 written in rectangular form has 𝑥- and 𝑦-components as shown. These will be equal to the displacement from the origin as shown in the graph. Vector 𝐕 can also be written in polar form 𝑟, 𝜃, where 𝑟 is the magnitude or length of the vector and 𝜃 is the angle that the vector makes with the positive 𝑥-axis.
In this question, 𝑟 is equal to six centimeters and 𝜃 is equal to 60 degrees. We can convert from polar form to rectangular form using our knowledge of right angle trigonometry such that 𝑥 is equal to 𝑟 cos 𝜃 and 𝑦 is equal to 𝑟 sin 𝜃. Substituting in our values for 𝑟 and 𝜃, we have 𝑥 is equal to six multiplied by cos of 60 degrees. And since cos of 60 degrees is one-half, 𝑥 is equal to six multiplied by one-half, which is equal to three.
In the same way, we have 𝑦 is equal to six multiplied by sin of 60 degrees. Since the sin of 60 degrees is root three over two, 𝑦 is equal to six multiplied by root three over two. And this is equal to three root three. Vector 𝐀 is therefore equal to three, three root three. And the correct answer from the four listed is option (B).