Question
Find the angle between vector u=3i+\sqrt(3)j and vector v=-2i-5j to the nearest degree.
A. 82
B. 38
C. 142
D. 98
A. 82
B. 38
C. 142
D. 98
Answers
GPT 3.5
To find the angle between two vectors, we can use the formula: cos(theta) = (u.v) / (||u|| * ||v||), where u.v represents the dot product of the two vectors, ||u|| represents the magnitude of vector u, and ||v|| represents the magnitude of vector v.
First, let's calculate the dot product of u and v:
u.v = (3)(-2) + (√3)(-5) = -6 - 5√3
Next, let's calculate the magnitudes of vectors u and v:
||u|| = sqrt((3)^2 + (√3)^2) = sqrt(9 + 3) = sqrt(12) = 2√3
||v|| = sqrt((-2)^2 + (-5)^2) = sqrt(4 + 25) = sqrt(29)
Now, we can plug these values into the formula to find cos(theta):
cos(theta) = (-6 - 5√3) / (2√3 * √29) = (-6 - 5√3) / (2√87)
Using a calculator, we find cos(theta) is approximately equal to -0.8714. Therefore, theta is approximately equal to 140.44 degrees.
Rounded to the nearest degree, the angle between vector u and vector v is 140 degrees. Therefore, the answer is not within the options provided.
First, let's calculate the dot product of u and v:
u.v = (3)(-2) + (√3)(-5) = -6 - 5√3
Next, let's calculate the magnitudes of vectors u and v:
||u|| = sqrt((3)^2 + (√3)^2) = sqrt(9 + 3) = sqrt(12) = 2√3
||v|| = sqrt((-2)^2 + (-5)^2) = sqrt(4 + 25) = sqrt(29)
Now, we can plug these values into the formula to find cos(theta):
cos(theta) = (-6 - 5√3) / (2√3 * √29) = (-6 - 5√3) / (2√87)
Using a calculator, we find cos(theta) is approximately equal to -0.8714. Therefore, theta is approximately equal to 140.44 degrees.
Rounded to the nearest degree, the angle between vector u and vector v is 140 degrees. Therefore, the answer is not within the options provided.