SOLUTION: Write the following vector in standard form. -4,-390°

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Question 1206661: Write the following vector in standard form. -4,-390°
Found 2 solutions by Edwin McCravy, math_tutor2020:
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
Draw the vector. Remember that negative angles are measured rotating
CLOCKWISE from the right side of the x-axis. And the length, or magnitude
of the vector is 4. 



Draw a perpendicular from the tip of the vector to the x-axis:


matrix%281%2C3%2Cx=r%2Acos%28theta%29%2C+%22%2C%22%2Cy=r%2Asin%28theta%29%29
matrix%281%2C3%2Cx=4%2Acos%28-390%5Eo%29%2C+%22%2C%22%2C+y=4%2Asin%28-390%5Eo%29%29

To get an equivalent positive angle add 360 twice, or 720 to the angle,
getting -390+360+360 = 330 and the reference angle is 30

matrix%281%2C3%2Cx=4%2Acos%28330%5Eo%29%2C+%22%2C%22%2C+y=4%2Asin%28330%5Eo%29%29

matrix%281%2C3%2Cx=4%2A%28sqrt%283%29%2F2%29%2C+%22%2C%22%2C+y=4%2A%28-1%2F2%29%29

matrix%281%2C3%2Cx=2sqrt%283%29%2C+%22%2C%22%2C+y=-2%29

The horizontal-component is the x-value, a multiple of the unit vector i.
 
The vertical-component is the y-value, a multiple of the unit vector j.

Answer: 2sqrt%283%29%2Ai%2B-2%2Aj

Edwin

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Place yourself at the origin. Face directly east.

Rotate 360 degrees clockwise and you'll still face east.
The clockwise rotation is due to theta being negative.
Rotate another 30 degrees clockwise to aim along the vector that tutor Edwin has drawn.
This is in quadrant 4 which is the southeast quadrant.
Note how angles -390 and -30 are coterminal.

If r > 0 was the case, then we'd walk from the origin 4 units in the direction vector Edwin has shown.
However, because r < 0 instead, we must walk 4 units in the complete opposite direction.
Think "negative = opposite".

This will mean the answer vector should point in the northwest quadrant (aka quadrant 2).
This is when x < 0 and y > 0.

r = -4
theta = -390

x = r*cos(theta) and y = r*sin(theta)
x = -4*cos(-390) and y = -4*sin(-390)
x = -4*cos(-30) and y = -4*sin(-30)
x = -4*cos(30) and y = 4*sin(30)
x = -4*( sqrt(3)/2 ) and y = 4*(1/2)
x = -2*sqrt(3) and y = 2

The answer in < x, y > form would be < -2*sqrt(3), 2 >

This would be equivalent to writing -2*sqrt(3)*i + 2j where i,j are the unit vectors pointing along the positive x axis and y axis respectively.
i = <1,0>
j = <0,1>

Edwin has the right idea, but needs to do a sign flip for the x and y components.