Answer:
|v| = 1
θ = 240°
Explanation:
You want the magnitude and angle of the vector with components (-1/2, -√3/2).
Magnitude
The length of the vector can be found using the Pythagorean theorem, or the distance formula:
|v| = √((-1/2)² +(-√3/2)²) = √(1/4 +3/4) = √1
|v| = 1
Angle
The angle can be found using the tangent relation, considering quadrant. The negative coordinate values tell you this is a third quadrant angle, so its measure will be 180° added to the first-quadrant result from the arctan function.
θ = arctan(y/x) = arctan((-√3/2)/(-1/2)) = arctan(√3) = 60° +180°
θ = 240°
Alternate approaches
The coordinates of a point on the unit circle are (cos(θ), sin(θ)). You probably recognize 1/2 = cos(60°), and √3/2 = sin(60°). This tells you two things:
- the vector is to a point on the unit circle, so |v| = 1
- the reference angle is 60°. θ is 3rd quadrant, so is 240°.
If you use your calculator to find the angle associated with artan(√3), you can also use your calculator to convert to polar coordinates. This is shown in the attachment. Of course, an angle of -120° is the same as +240°.
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