Vector with Trig Function Math Question
I am reading a book on electromagnetic fields and in one of the examples describing vector analysis, the text reads, "express the vector velocity v = av 55 (ms-1) at the point P(0,2,5), shown in Fig. 1-4 to be in the y-z plane, in component form, using unit vectors."
So the text goes on, "The vector velocity v can be expressed as v = xvx + yvy + zvz. From Fig. 1-4, vx = 0, vy = 55 sin β, vz = -55 cos β, cos β = 4/5, and sin β = 3/5."
I guess I have a couple of questions, all related:
First, how did the authors derive a vector originating from point P(0, 2, 5) to 4 units down the z-axis and 3 units right on the y-axis? How do they derive this graphic depiction from the original information presented? What is the meaning of (ms-1)?
Second, how is the vector component, vy, equal to 55 sin β?
I understand sin is a function of the ratio of the opposite side of the angle to the hypotenuse, and so the sin of β will render 3/5 from the depiction, but how did they get that in the first place? Also, why is it 55 multiplied by this ratio?
NOTE: I am new to this forum and I have not expressed these formulas with the corresponding "hats" or "carets" or vector bars in their notation.
Re: Vector with Trig Function Math Question
It is basically as you have pointed out to do with Pythagoras' theorem.
We know that sin(theta) = opposite/hypotenuse and hypotenuse = 55 so opposite = 55*sin(theta) = vy
Also 55*cos(theta) = vx using a similar argument.
It's just a matter of identifying the right angles and sides of the triangle.