Find two unit vectors that make an angle of 60° with
v =
<6, 8>
sketch it first
angle from x axis = tan^-1 (8/6) = 53.1 deg
so first clockwise 60 degrees
so -6.87 degrees (in quadrant 4)
tan -6.67 = -.12
so if x = 1, y = -.12
now we want that to be a unit vector so find magnitude of what we have
r = sqrt(x^2 + y^2)
= sqrt 1.014 = 1.007
so
< 1/1.007 , -.12/1.007 >
now you can do the 60 degrees counterclockwise from 53.1 which will be in quadrant 2, x negative, y positive
how do you do it without using a calculator?
To find two unit vectors that make an angle of 60° with a given vector v = <6, 8>, we can follow these steps:
Step 1: Normalize the vector v by dividing each component by its magnitude.
The magnitude of v = |v| = √(6^2 + 8^2) = √(36 + 64) = √100 = 10.
Normalized vector v = v / |v| = <6/10, 8/10> = <0.6, 0.8>.
Step 2: Rotate the normalized vector v by 60° in both clockwise and counterclockwise directions to get the two unit vectors.
Counterclockwise direction:
To rotate the vector counterclockwise, we can use the rotation matrix formula:
[cosθ -sinθ]
[sinθ cosθ]
For a counterclockwise rotation of 60°, the rotation matrix is:
[cos60° -sin60°] = [1/2 -√3/2]
[sin60° cos60°] = [√3/2 1/2]
Multiply the rotation matrix by the normalized vector v:
<0.6 * 1/2 - 0.8 * √3/2, 0.6 * √3/2 + 0.8 * 1/2> = <0.3 - 0.8√3, 0.3√3 + 0.4>
Thus, one unit vector that makes a 60° angle counterclockwise with v is approximately <0.3 - 0.8√3, 0.3√3 + 0.4>.
Clockwise direction:
To rotate the vector clockwise, we use the same rotation matrix but with a negative angle (-60°).
Multiply the rotation matrix by the normalized vector v:
<0.6 * 1/2 + 0.8 * √3/2, -0.6 * √3/2 + 0.8 * 1/2> = <0.3 + 0.8√3, -0.3√3 + 0.4>
Thus, one unit vector that makes a 60° angle clockwise with v is approximately <0.3 + 0.8√3, -0.3√3 + 0.4>.
Therefore, the two unit vectors that make a 60° angle with v = <6, 8> are approximately:
<0.3 - 0.8√3, 0.3√3 + 0.4> and <0.3 + 0.8√3, -0.3√3 + 0.4>.