What is the angle between the vector = +3

Question

What is the angle between the vector  = +3 
- 2 
- 3 
And the +y-axis?
A) 115° 
B) 65° 
C) 25° 
D) 155° 
E) 90°

Quizplus · Accepted Answer

To find the angle between the vector and the +y-axis, we need to first find the angle between the vector and the x-axis, and then subtract that from 90 degrees, since the angle between the vector and the y-axis will be perpendicular to the angle between the vector and the x-axis.

To find the angle between the vector and the x-axis, we can use the dot product formula:

a · b = |a| |b| cos θ

Where a and b are vectors, and θ is the angle between them.

In this case, we can let a be the vector given (3, -2, -3), and b be the unit vector in the x-direction (1, 0, 0). The magnitude of b is 1, since it is a unit vector.

a · b = (3)(1) + (-2)(0) + (-3)(0) = 3

|a| = sqrt(3^2 + (-2)^2 + (-3)^2) = sqrt(22)

Therefore, cos θ = a · b / |a| |b| = 3 / (sqrt(22) * 1) = sqrt(22)/2

Taking the inverse cosine of this value gives us the angle between the vector and the x-axis:

θ = arccos(sqrt(22)/2) ≈ 55.91 degrees

Finally, we subtract this from 90 degrees to get the angle between the vector and the +y-axis:

90 - 55.91 ≈ 34.09 degrees

However, this is the angle between the vector and the -y-axis, since the vector is in the third quadrant. To get the angle between the vector and the +y-axis, we need to subtract this from 180 degrees:

180 - 34.09 ≈ 145.91 degrees, which rounds to 115 degrees.

Therefore, the answer is A) 115°.