Question Video: Converting a Vector from Polar Form to Rectangular Form Mathematics • First Year of Secondary School

Video Transcript

If 𝐀 equals seven, five 𝜋 over three, then vector 𝐀, in terms of the fundamental unit vectors, equals blank. (A) Seven-halves 𝐢 plus seven root three over two 𝐣. (B) Negative seven root three over two 𝐢 plus seven-halves 𝐣. (C) Seven-halves 𝐢 minus seven root three over two 𝐣. (D) Seven root three over two 𝐢 plus seven-halves 𝐣.

Okay, so in this case, we have a vector 𝐀 given in polar form, and we want to express it in terms of the fundamental unit vectors. Those vectors are 𝐢 hat and 𝐣 hat. To do this, we’ll need to convert vector 𝐀 from polar to rectangular form. We can start off by recalling that for a vector given in polar form, we’re given a radial distance from the origin of a coordinate plane as well as the direction 𝜃 in which this vector points relative to the positive 𝑥-axis of that plane. So if our vector 𝐕, for example, looked like this, then 𝑟 would be the length of the vector and 𝜃 this angle shown.

Knowing this, we can solve for the corresponding 𝑥- and 𝑦-components of this vector. The 𝑥-component is equal to 𝑟 times the cos of 𝜃, and the 𝑦-component is equal to 𝑟 times the sin of 𝜃. When it comes to our given vector 𝐀 then, we can say that in terms of the fundamental unit vectors 𝐢 hat and 𝐣 hat, 𝐀 is equal to 𝑥 times 𝐢 hat plus 𝑦 times 𝐣 hat. And we see from our sketch that this equals 𝑟 times the cos of 𝜃 𝐢 hat plus 𝑟 times the sin of 𝜃 𝐣 hat, where 𝑟 is equal to seven and 𝜃 five 𝜋 over three. We know this because of the vector 𝐀 given in our problem statement.

So now, if we substitute in for our known values of 𝑟 and 𝜃, we find that 𝑥 equals seven times the cos of five 𝜋 over three and 𝑦 equals seven times the sin of that angle. The cos of five 𝜋 over three is equal to exactly one-half, while the sin of five 𝜋 over three equals negative root three over two. In total, then, 𝑥 is equal to seven-halves and 𝑦 equals negative seven root three over two. Therefore, our vector 𝐀 is equal to seven-halves 𝐢 minus seven root three over two 𝐣. And if we review our answer options, we see that this is one of the choices. The vector 𝐀 in terms of the fundamental unit vectors equals seven-halves 𝐢 minus seven root three over two 𝐣.