Fill in the blank. If the magnitude of vector 𝐀 is equal to four centimeters, then vector 𝐀 equals what.
In this question, we want to give the component or rectangular form of vector 𝐀, given a graphical representation together with the magnitude of the vector. We are told that this magnitude or length is equal to four centimeters. The 𝑥- and 𝑦-components of the vector can be found from the diagram as shown. And since the magnitude of a vector is its length, we have a right triangle with two unknown side lengths.
We can therefore calculate the values of 𝑥 and 𝑦 using our knowledge of right angle trigonometry. The sine ratio tells us that sin 𝜃 is equal to the opposite over the hypotenuse. Substituting in the values from the diagram, we have sin of 30 degrees is equal to 𝑦 over four. Using our knowledge of special angles, we know that sin of 30 degrees is equal to one-half. We can then multiply through by four, giving us 𝑦 is equal to two. The cosine ratio tells us that cos 𝜃 is equal to the adjacent over the hypotenuse. This means that cos of 30 degrees is equal to 𝑥 over four. Once again, using our knowledge of special angles, we know that cos of 30 degrees is equal to root three over two. And multiplying through by four, 𝑥 is equal to two root three. We now have values of 𝑥 and 𝑦 as required such that vector 𝐀 written in rectangular form is two root three, two.
An alternative method would’ve been to have considered the vector in polar form first. This is written 𝑟𝜃, where 𝑟 is the magnitude or length of the vector and 𝜃 is the angle the vector makes with the positive 𝑥-axis, usually written in radians. We were told in the question that the magnitude of vector 𝐀 was four centimeters, so 𝑟 is equal to four. Since 180 degrees is equal to 𝜋 radians and 180 divided by six is equal to 30, then 30 degrees is equal to 𝜋 over six radians. And this means that 𝜃 is equal to 𝜋 over six.
We can then convert vector 𝐀 from polar form to rectangular form using the fact that 𝑥 is equal to 𝑟 multiplied by cos 𝜃 and 𝑦 is equal to 𝑟 multiplied by sin 𝜃. Substituting in our values, we have 𝑥 is equal to four multiplied by cos of 𝜋 over six and 𝑦 is equal to four multiplied by sin of 𝜋 over six. This once again gives us values of 𝑥 and 𝑦 of two root three and two, respectively, proving that vector 𝐀 is indeed equal to two root three, two.