Angle Between two surfaces and a point

In summary, the angle between two surfaces at a point of intersection can be found by calculating the angle between their normal vectors at that point. In order to do this, the surfaces should be written in the form of f(x,y,z)= constant and the gradients of these functions should be found. Then, the formula \vec{u}\cdot\vec{v}= |\vec{u}||\vec{v}|cos(\theta) can be used to determine the angle between the two vectors. While this question was posted in the Precalculus category, it requires the use of Calculus.
  • #1
CompStang
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Find the angle between the surfaces defined by r^2= 9 and x + y + z^2= 1 at the point (2,-2,1)? --- I know this should be extremely simple but it is blowing my mind for some reason. Any help would be greatly appreciated.
 
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  • #2
The angle between two surfaces at a point pf intersection is the angle between their normal vectors at that point. And if you write the surfaces as f(x,y,z)= constant, the normal vector is in the same direction as [itex]\nabla f[/itex]. You probably want to write the first surface in Cartesian coordinates as x2+ y2+ z2= 9. Find the gradients of those two functions and then use the fact that
[itex]\vec{u}\cdot\vec{v}= |\vec{u}||\vec{v}|cos(\theta)[/itex] where [itex]\theta[/itex] is the angle between the two vectors.

Although this was posted in Precalculus, I don't believe this can be done without using Calculus.
 
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  • #3


The angle between two surfaces at a given point can be found by calculating the dot product of the surface normal vectors at that point. In this case, the surface normal vector for r^2=9 is <2x, 2y, 2z>, and the surface normal vector for x+y+z^2=1 is <1, 1, 2z>.

At the point (2,-2,1), the normal vectors are <4, -4, 2> and <1, 1, 2>. Using the dot product formula, we get:

<4, -4, 2> • <1, 1, 2> = (4)(1) + (-4)(1) + (2)(2) = 4 + (-4) + 4 = 4

The angle between the two surfaces at this point is given by the formula cosθ = (a • b) / (|a||b|), where a and b are the normal vectors and |a| and |b| are their respective magnitudes.

In this case, |a| = √(4^2 + (-4)^2 + 2^2) = √36 = 6 and |b| = √(1^2 + 1^2 + 2^2) = √6.

Therefore, cosθ = (4) / (6√6) = 2 / (3√6). Taking the inverse cosine, we get θ = 41.41 degrees.

So, the angle between the two surfaces at the point (2,-2,1) is approximately 41.41 degrees.
 

1. What is the angle between two surfaces?

The angle between two surfaces is the measurement of the amount of rotation or deviation between the two surfaces. It is the smallest angle formed between the two surfaces at their point of intersection.

2. How is the angle between two surfaces calculated?

The angle between two surfaces can be calculated using trigonometric functions such as sine, cosine, and tangent. The specific formula used will depend on the type of surfaces and the information available about their orientation.

3. What is the significance of the angle between two surfaces?

The angle between two surfaces is important in determining the relationship between the surfaces, such as whether they are parallel, perpendicular, or at an angle to each other. It is also crucial in fields such as engineering, architecture, and physics for calculating forces and stresses on structures.

4. Can the angle between two surfaces change?

Yes, the angle between two surfaces can change if the orientation or position of either surface is altered. For example, if one surface is rotated or translated, the angle between it and the other surface will change.

5. How does the angle between two surfaces affect the point at which they intersect?

The angle between two surfaces can determine the location and direction of the point at which they intersect. For example, if the angle between two surfaces is 90 degrees, their intersection point will be perpendicular to both surfaces. If the angle is 0 degrees, the surfaces will intersect at a single point.

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