Question Video: Converting a Vector from Polar Form to Rectangular Form Mathematics

If 𝐀 = 〈7, 5πœ‹/3βŒͺ, then vector 𝐀, in terms of the fundamental unit vectors, equals οΌΏ.

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Video Transcript

If 𝐀 equals seven, five πœ‹ over three, then vector 𝐀, in terms of the fundamental unit vectors, equals blank. (A) Seven-halves 𝐒 plus seven root three over two 𝐣. (B) Negative seven root three over two 𝐒 plus seven-halves 𝐣. (C) Seven-halves 𝐒 minus seven root three over two 𝐣. (D) Seven root three over two 𝐒 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 𝐒 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 πœƒ in which this vector points relative to the positive π‘₯-axis of that plane. So if our vector 𝐕, for example, looked like this, then π‘Ÿ would be the length of the vector and πœƒ this angle shown.

Knowing this, we can solve for the corresponding π‘₯- and 𝑦-components of this vector. The π‘₯-component is equal to π‘Ÿ times the cos of πœƒ, and the 𝑦-component is equal to π‘Ÿ times the sin of πœƒ. When it comes to our given vector 𝐀 then, we can say that in terms of the fundamental unit vectors 𝐒 hat and 𝐣 hat, 𝐀 is equal to π‘₯ times 𝐒 hat plus 𝑦 times 𝐣 hat. And we see from our sketch that this equals π‘Ÿ times the cos of πœƒ 𝐒 hat plus π‘Ÿ times the sin of πœƒ 𝐣 hat, where π‘Ÿ is equal to seven and πœƒ five πœ‹ 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 π‘Ÿ and πœƒ, we find that π‘₯ equals seven times the cos of five πœ‹ over three and 𝑦 equals seven times the sin of that angle. The cos of five πœ‹ over three is equal to exactly one-half, while the sin of five πœ‹ over three equals negative root three over two. In total, then, π‘₯ is equal to seven-halves and 𝑦 equals negative seven root three over two. Therefore, our vector 𝐀 is equal to seven-halves 𝐒 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 𝐒 minus seven root three over two 𝐣.

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