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.