### 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, weβll need to convert
vector π from polar to rectangular form. We can start off by recalling that
for a vector given in polar form, weβre 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£.