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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 = a_{v} 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."

Attachment 26546

So the text goes on, "The vector velocity v can be expressed as v = xv_{x} + yv_{y} + zv_{z}. From Fig. 1-4, v_{x} = 0, v_{y} = 55 sin β, v_{z} = -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, v_{y}, 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

Hey KClose1983.

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.