find the sine of the angle perpendicular to a plane given an x slope and a y slope?

[Inhahe]

Say I have a plane, and I know its tilt by an X slope and a Y slope, and I need to know the sine of the angle between a line perpendicular to the plane and the z axis... what's the most efficient way to do that? Thanks

 

[Inhahe]

I thought maybe sin(atan(sqrt((x_slope)^2+(y_slope)^2))), but I don't think that's right..

 

[LCKurtz]

I would think in terms of vectors. If the slopes in the [MATH]x[/MATH] and [MATH]y[/MATH] directions are [MATH]a[/MATH] and [MATH]b[/MATH], respectively, then the equation of the plane can be written as [MATH]z = ax +by + c[/MATH], or [MATH]ax +by -z = c[/MATH]. So the normal to the plane is [MATH]\vec N = \langle a, b, -1\rangle[/MATH]. A vector along the [MATH]z[/MATH] axis is [MATH]\langle 0,0,1\rangle[/MATH]. Now the cross product of those two involves the sine of the angle you seek. Do you know that formula? Can you take it from there?

 

[Inhahe]

Is this right: sqrt(a^2+b^2+1)? that's what wolfram alpha gives me. https://www.wolframalpha.com/input/?i=⟨+a+,+b+,+−+1+⟩+⨉+⟨+0+,+0+,+1+⟩

 

[LCKurtz]

Is this right: sqrt(a^2+b^2+1)? that's what wolfram alpha gives me. https://www.wolframalpha.com/input/?i=⟨+a+,+b+,+−+1+⟩+⨉+⟨+0+,+0+,+1+⟩

The sine of an angle between [MATH]0^\circ[/MATH] and [MATH]180^\circ[/MATH] is a number in the interval [MATH][0,1][/MATH]. That kind of rules out that answer eh? Why don't you try actually working it out and show us what you get?

 

[Inhahe]

I found this: https://www.mathsisfun.com/algebra/vectors-cross-product.html

According to that, the cross product of <a, b, -1> and <0, 0, 1> would be, let's see..

<b, a, 0> if i calculated correctly.

but that's not a value between 0 and 1, and it's not even a single value..

 

[LCKurtz]

In post #3 I asked you if you know the formula involving cross product and the sine function. You haven't answered that question. Please show us the formula that involves cross product and [MATH]\sin\theta[/MATH]. What is the formula for [MATH]\sin\theta[/MATH] that you are trying to use? Show us. Look it up if you don't know it.

 

[Inhahe]

I don't know the formula for cross product. I looked it up and found that site I linked to. https://www.mathsisfun.com/algebra/vectors-cross-product.html. The part of that page I used is this:

Example: The cross product of a = (2,3,4) and b = (5,6,7)

• cx = aybz − azby = 3×7 − 4×6 = −3
• cy = azbx − axbz = 4×5 − 2×7 = 6
• cz = axby − aybx = 2×6 − 3×5 = −3

Answer: a × b = (−3,6,−3)

As for sin θ, it seemed like you were saying that the cross product of <a, b, -1> and <0, 0, 1> would *give me* the sine of θ, maybe I misunderstood. On closer inspection, I don't know what you mean by "the cross product involves θ" or "formula that involves cross product and sin θ." I can look up a formula for cross product, and I can look up a formula for sine (or better, I can use just the sin function), but I'm not sure what to look up to understand how one "involves" the other. I think I would have to study a lot of trig to understand this. I was hoping for a quick answer. Thanks.

 

[LCKurtz]

It's hard to solve problems using material you haven't learned. If you haven't learned anything about vector methods it would be good to explain that when you are advised to use them. We aren't mind readers. How do you expect to calculate [MATH]\sin\theta[/MATH] using vector methods if you don't know a formula to use? The things you should know about cross product are how to calculate it and what it represents. So, given a couple of vectors:
1. How do you calculate their cross product? You seem to have found how to do that.
2. What does it represent? A vector normal to the plane of the originals in the direction given by the right hand rule.
That gives the direction, the length is given by [MATH]|\vec a \times \vec b| = |\vec a||\vec b|\sin\theta[/MATH], where [MATH]\theta[/MATH] is the angle between the original vectors, where [MATH]|\vec a|[/MATH] represents the length of [MATH]\vec a[/MATH].

That last equation "involves" [MATH]\sin\theta[/MATH] and you should be able to figure out its value using the information you are given.
[Edit, added] Note that the web link you referenced about cross products tells you all the above information if you read it carefully.