Question Video: Converting a Vector from Polar Form to Rectangular Form | Nagwa Question Video: Converting a Vector from Polar Form to Rectangular Form | Nagwa

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

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If π = β©7, 5π/3βͺ, then vector π, in terms of the fundamental unit vectors, equals οΌΏ.

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Video Transcript

If π equals seven, five π over three, then vector π, in terms of the fundamental unit vectors, equals blank. (A) Seven-halves π’ plus seven root three over two π£. (B) Negative seven root three over two π’ plus seven-halves π£. (C) Seven-halves π’ minus seven root three over two π£. (D) Seven root three over two π’ plus seven-halves π£.

Okay, so in this case, we have a vector π given in polar form, and we want to express it in terms of the fundamental unit vectors. Those vectors are π’ hat and π£ hat. To do this, weβll need to convert vector π from polar to rectangular form. We can start off by recalling that for a vector given in polar form, weβre given a radial distance from the origin of a coordinate plane as well as the direction π in which this vector points relative to the positive π₯-axis of that plane. So if our vector π, for example, looked like this, then π would be the length of the vector and π this angle shown.

Knowing this, we can solve for the corresponding π₯- and π¦-components of this vector. The π₯-component is equal to π times the cos of π, and the π¦-component is equal to π times the sin of π. When it comes to our given vector π then, we can say that in terms of the fundamental unit vectors π’ hat and π£ hat, π is equal to π₯ times π’ hat plus π¦ times π£ hat. And we see from our sketch that this equals π times the cos of π π’ hat plus π times the sin of π π£ hat, where π is equal to seven and π five π over three. We know this because of the vector π given in our problem statement.

So now, if we substitute in for our known values of π and π, we find that π₯ equals seven times the cos of five π over three and π¦ 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, π₯ is equal to seven-halves and π¦ equals negative seven root three over two. Therefore, our vector π is equal to seven-halves π’ minus seven root three over two π£. And if we review our answer options, we see that this is one of the choices. The vector π in terms of the fundamental unit vectors equals seven-halves π’ minus seven root three over two π£.

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