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hkor
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how do i find the angle between the plane x=0 and the plane 2x+3y-z=4?
hkor said:how do i find the angle between the plane x=0 and the plane 2x+3y-z=4?
hkor said:Oh great thanks for the hint. I wasn't aware that for the angle between intersecting planes, the normal vector was used, unlike in the angle between intersecting lines. I think this is correct - the normal vector of 2x+3y-z=4 is (2,3,-1) and the normal vector of x=0 is (1). Thus the dot product would be (2)(1)+(3)+(-1) = 4. The magnitude would be sqrt(2^2+3^2+(-1)^2) * sqrt(1)= sqrt(14). thus the angle is equal to cos^-1 (4/sqrt(14))= cos^-1(1.069). I think this is correct however when typing the formula for the angle into my calculator a math error comes up. Can you explain this?
As chiro said, you need to use the unit normal. Divide by [itex]\sqrt{4+ 9+ 1}= \sqrt{14}[/itex]hkor said:Oh great thanks for the hint. I wasn't aware that for the angle between intersecting planes, the normal vector was used, unlike in the angle between intersecting lines. I think this is correct - the normal vector of 2x+3y-z=4 is (2,3,-1)
That's not even a vector! The normal vector to the yz-plane, x= 0, is (1, 0, 0), which is unit length.and the normal vector of x=0 is (1).
You do understand that (3) is the same as (3)(1), don't you? You are taking the dot product of (2, 3, -1) and (1, 1, 1). You want, instead, (2, 3, -1).(1, 0, 0)= 2. Remember to divide by [itex]\sqrt{14}[/itex].Thus the dot product would be (2)(1)+(3)+(-1) = 4.
magnitude would be sqrt(2^2+3^2+(-1)^2) * sqrt(1)= sqrt(14). thus the angle is equal to cos^-1 (4/sqrt(14))= cos^-1(1.069). I think this is correct however when typing the formula for the angle into my calculator a math error comes up. Can you explain this?
hkor said:So when finding the dot product i use the unit vector (2/sqrt(14),3/sqrt(14),-1/sqrt(14)) . (1,0,0) = 2/sqrt(14). The magnitude of the normal vectors will be one, thus the ans will be... cos ^-1 (2/sqrt(14))= 1.0069?
The angle between two planes is the measure of the angle formed by the intersection of the two planes. It is the smallest angle between two lines, one on each plane, drawn perpendicular to the intersection line.
To calculate the angle between two planes, you can use the dot product formula: angle = cos⁻¹ (|a∙b| / |a| ∙ |b|), where a and b are the normal vectors of the two planes. Alternatively, you can use the cross product formula: angle = sin⁻¹ (|a x b| / |a| ∙ |b|).
Yes, the angle between two planes can be greater than 90 degrees. It can range from 0 degrees (when the planes are parallel) to 180 degrees (when the planes are perpendicular).
If the angle between two planes is 0 degrees, it means that the two planes are parallel. This means that they do not intersect and have the same slope in all directions.
The concept of the angle between two planes is important in fields such as engineering, physics, and architecture. It is used to measure the orientation and relationship between two surfaces, which can be useful in designing and constructing buildings, bridges, and other structures. It is also used in navigation and 3D modeling.