Video Transcript
Which of the following is the polar representation of vector 𝐀 which is equal to two root three, two?
In this question, vector 𝐀 is given in rectangular, or component, form. We know that any vector 𝐕 can be written in this form 𝑥, 𝑦, where 𝑥 and 𝑦 are the displacements from the origin in the 𝑥- and 𝑦-direction, respectively. 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 the vector makes with the positive 𝑥-axis.
In the five sketches, we’re given the magnitude of the vector, and we are also given the angle the vector makes with the positive 𝑥-axis. Since the vector lies in the first quadrant, we can determine the polar form using the fact that 𝑟 is equal to the square root of 𝑥 squared plus 𝑦 squared and 𝜃 is equal to the inverse tan of 𝑦 over 𝑥. The 𝑥-component of our vector is two root three, and the 𝑦-component is two. Therefore, 𝑟 is equal to the square root of two root three squared plus two squared. The right-hand side simplifies to the square root of 12 plus four. And this is equal to root 16. Since 𝑟 must be positive, 𝑟 is equal to four. And this rules out options (C), (D), and (E).
Moving on to the angle, we have 𝜃 is equal to the inverse tan of two over two root three. Both the numerator and denominator are divisible by two. So 𝜃 is equal to the inverse tan of one over root three. Ensuring that our calculator is in degree mode, typing this in gives us 𝜃 is equal to 30 degrees.
The correct answer is therefore option (A), as this shows a vector with magnitude, or length, equal to four and an angle 𝜃 between the vector and the positive 𝑥-axis of 30 degrees.
