### Video Transcript

Which of the following is the polar representation of vector π which is equal to two root three, two?

In this question, vector π is given in rectangular, or component, form. We know that any vector π can be written in this form π₯, π¦, where π₯ and π¦ are the displacements from the origin in the π₯- and π¦-direction, respectively. Vector π can also be written in polar form π, π, where π is the magnitude, or length, of the vector and π is the angle the vector makes with the positive π₯-axis.

In the five sketches, weβre given the magnitude of the vector, and we are also given the angle the vector makes with the positive π₯-axis. Since the vector lies in the first quadrant, we can determine the polar form using the fact that π is equal to the square root of π₯ squared plus π¦ squared and π is equal to the inverse tan of π¦ over π₯. The π₯-component of our vector is two root three, and the π¦-component is two. Therefore, π 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 π must be positive, π is equal to four. And this rules out options (C), (D), and (E).

Moving on to the angle, we have π is equal to the inverse tan of two over two root three. Both the numerator and denominator are divisible by two. So π is equal to the inverse tan of one over root three. Ensuring that our calculator is in degree mode, typing this in gives us π 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 π between the vector and the positive π₯-axis of 30 degrees.