# Question Video: Representing Vectors Geometrically Mathematics

Fill in the blank: If vector 𝐏𝐌 = <4√3, 4>, then the polar form of vector 𝐏𝐌 is ＿.

03:46

### Video Transcript

Fill in the blank. If vector 𝐏𝐌 is equal to four root three four, then the polar form of vector 𝐏𝐌 is what.

In this question, we are given vector 𝐏𝐌 in rectangular form and are asked to represent it in polar form. We begin by recalling that any vector 𝐕 written in component or rectangular form 𝑥, 𝑦 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 from the positive 𝑥-axis. We can convert from one form to the other using the fact that 𝑥 is equal to 𝑟 multiplied by cos 𝜃 and 𝑦 is equal to 𝑟 multiplied by sin 𝜃. We could also represent this graphically by drawing the vector on the two-dimensional coordinate plane.

In this question, vector 𝐏𝐌 is equal to four root three four. We can then create a right triangle on our diagram to calculate the values of 𝑟 and 𝜃. The Pythagorean theorem states that in any right triangle, 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the hypotenuse and 𝑎 and 𝑏 are the lengths of the shorter sides of the triangle. Substituting in the lengths of the two shorter sides of our triangle, we have four root three squared plus four squared is equal to 𝑟 squared. Four root three squared is equal to 48 and four squared is 16. So we have 𝑟 squared is equal to 48 plus 16, which is equal to 64. We can then square root both sides of this equation. And since 𝑟 must be positive, we have 𝑟 is equal to eight.

Next, we can calculate the value of 𝜃 using our knowledge of right angle trigonometry. Since tan 𝜃 is equal to the opposite over the adjacent, this is equal to four over four root three, which simplifies to one over root three. We can then take the inverse tangent of both sides such that 𝜃 is equal to the inverse tan of one over root three. And since 𝜃 lies in the first quadrant, this is equal to 30 degrees or 𝜋 over six radians.

We now have values for both 𝑟 and 𝜃 such that the polar form of vector 𝐏𝐌 is eight 𝜋 over six. It is worth noting that had we used the fact that 𝑥 is equal to 𝑟 cos 𝜃 and 𝑦 is equal to 𝑟 sin 𝜃, instead of drawing the diagram, we can simply quote the results that 𝑟 squared is equal to 𝑥 squared plus 𝑦 squared and tan 𝜃 is equal to 𝑦 over 𝑥. This would’ve given us the same equations as from the diagram. And this method is often quicker in order to solve problems of this type.