Vector ~B has x, y, and z components of 3.3,2.7, and 7.7 units, respectively. Calculate the magnitude of ~B. What is the angle in degrees between ~B and the x-axis?
In the context of your question, the vector B points from the origin to the point (3.3,2.7,7.7). We would represent B as [3.3,2.7,7.7]. To determine the length of a vector, take it's norm: ||[x,y,z]|| = sqrt(x² + y² + z²)
so || B || = sqrt(3.3² + 2.7² + 7.7²) = 8.8 units
Any vector representing the x axis take the form [x 0 0]. Lets use the unit vector: X = [1,0,0], since it's norm is 1 (|| X || = 1). The angle between any two vectors u and v is:
cos(Θ) = u·v/(||u|| ||v||)
The dot product of u and v is simple since only the x-component is non-zero: u·v = 3.3*1 = 3.3
so cos(Θ) = 3.3 / 8.8*1 = .375
Θ = acos(.375) = 68°