Question Video: Representing Vectors Geometrically Mathematics

If |𝐀| = 6 cm, then 𝐀 = _.

02:27

Video Transcript

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).

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.