How did we get to the formula of x and y components of a vector?

[shivajikobardan]

I studied computer engineering and there were four mathematics subjects. But mostly they taught us to become like a calculator instead of focus on analytical thinking. So I am relearning part-time.

Imagine a situation like the one shown in Figure 6.5
Picture taken from the book: Computer Graphics by Peter Shirley et al

Based on my level of math knowledge, I got different answer.
I would do (180-(alpha+phi)) and calculate the cosine and sine values. Doing that, I would coincidentally get same value for yb as provided in the textbook. However, I got negated value for xb as cos is negative in second quadrant (STAC rule of trigonometry).

I asked chat bots and they told me to memorize the formula of polar co-ordinates. Which I do not know came from where. I hope to seek some help here.

 

[Dr.Peterson]

I would do (180-(alpha+phi)) and calculate the cosine and sine values.

Please tell us why you would do that, rather than what they did, and why you would or would not expect to get the same result.

 

[shivajikobardan]

Because we get a right angle triangle. and we know cos(theta)=base/hypotenuse and sin(theta)=perpendicular/hypotenuse

 

[blamocur]

Because we get a right angle triangle. and we know cos(theta)=base/hypotenuse and sin(theta)=perpendicular/hypotenuse

I have no idea how this might answer post #2. Can you be more specific and post the details of your thinking?

 

[Dr.Peterson]

Because we get a right angle triangle. and we know cos(theta)=base/hypotenuse and sin(theta)=perpendicular/hypotenuse

The angle used to find the components is the angle \*\alpha+\theta\), which is in standard position (from the positive x-axis to b). You appear to be using the angle from the negative x-axis to b, which is not in standard position.

Perhaps you are thinking too much about acute angles in a right triangle, and not about the general definition of trig functions.

 

[Dr.Peterson]

Somehow I failed to check what I wrote; let's get it right:

The angle used to find the components is the angle [imath]\alpha+\theta[/imath], which is in standard position (from the positive x-axis to b). You appear to be using the angle from the negative x-axis to b, which is not in standard position.

Perhaps you are thinking too much about acute angles in a right triangle, and not about the general definition of trig functions.

 

[shivajikobardan]

maybe. polar coordinates concept claim that x=rcos(angle with x) and y=rsin(angle with x). That's all.

 

[Dr.Peterson]

maybe. polar coordinates concept claim that x=rcos(angle with x) and y=rsin(angle with x). That's all.

And you do understand that that's the signed angle with the positive x-axis, not the reference angle (acute angle with closer part of axis). Right?

 

[shivajikobardan]

nope i do not understand that much depth. i am just getting by.

 

[Dr.Peterson]

nope i do not understand that much depth. i am just getting by.

Then you'll need to give us a clue as to what you do understand, so we can start there.

If the book you referred to is the one you are trying to learn from (I presume I can't access it legally), you can find the definition of polar coordinates there, and see what I stated somehow. Maybe you need to find the first thing in the book that you don't fully understand, and ask about that.

Or start with a more basic book, perhaps one that is freely accessible online so it would be easier to discuss with others.

 

[blamocur]

Or start with a more basic book, perhaps one that is freely accessible online so it would be easier to discuss with others.

Or find a page online like this one.

P.S. I thought of recommending Wikipedia, but, IMHO, it is better as a reference source rather than a learning resource.