Angle Between Two Planes

Multivariable Calculus: Consider the planes x-y+z = 4 and 2x + y + z = 10. Determine whether the planes are parallel, perpendicular, or neither. If neither, find the cosine of the angle between them.

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[starlight]

Finally a math video on youtube that doesn't take 30 minutes to explain one concept.

[MathDoctorBob]

@antodez1234 You're welcome! There are two equivalent equations of the plane we use:

ax+by+cz=d and

a(x-x0) + b(y-y0) + c(z-z0) = 0.

The second one is better: it says take all vectors perpendicular to the vector (a, b, c) and based at the point (x0, y0, z0).

If we consider the angle between two planes, we are allowed to shift to parallel planes to get the same angle. If you shift both planes so they contain the origin, they now have equation in the form ax + by + cz = 0.
Hope it helps. - Bob

[MathDoctorBob]

You're welcome! You should think of the dot product as generalizing length of a vector. If we dot a vector with itself, we sum the squares of the components. In general, the dot product measures how much the vectors point in the same "direction, " and it does that by comparing components (by multiplying).

Think of R itself. If we multiply two one-vectors (numbers), a positive result means they have the same sign (direction), and negative means opposite. This also captures lengths.

[aNuthaRedneck]

Here's a problem/calculation I can't wrap my mind around. I am building an all metal house. The roofs have a slope of 4:12 and intersect at 90 degrees. I need to have a metal pan bent to fit in the valley of this intersection. What is the angle the metal valley should be bent? How does one calculate that angle? Or, what formula is used to calculate that angle?

[MathDoctorBob]

Yes. If you draw the picture, you'll note that the only change is the direction of one normal vector by -1 or 180 degrees.

[MathDoctorBob]

@jimmykim808 You're welcome! Thanks for the comment. - Bob

[alexandgarciacalle]

GreaVideo! thank you. But maybe I dont understand, why do you add the final products of the dot product if they are of different components? (i, j, k)
Thank you again.

[girl]

What if we arrive at cos aplha= -0.5.
We'll get the angle to be 120 degrees. But the question asks to find the 'acute' angle between the plane. What do I do now? Simply 180-120=60 ??

[cnthach]

awesome! explained it so much better than my professor =)!

[MathDoctorBob]

Your welcome, and good luck on finals!

[MathDoctorBob]

That's the idea. Here's what I get. First use cos(O) = (u, v)/|u||v| = -88/sqrt(115.168) =-.633109909. arccos gives the angle between 0 and pi, so O'=2.256360618 rad = 129.28 degrees. Going acute gives O=180-O'=51 degrees.

[luizpaganelli_]

Do we ignore the constants because the planes can be shifted anywhere and the acute angle upon intersection would still be the same?

[hansmalcolmringger]

good vid, but what's with the staff of wisdom?

[needsumlove]

Sir can u teach me if the answer is more than 90 degree, how should i know i should take the answer minus 90 or using 180- the ans.

[haydn]

chad explains how to find the angle between two planes

[virginialikesyou]

Thank you very much!!! Very helpful to meee!! <3

[MathDoctorBob]

@moosemooseson You're welcome! - Bob

[MathDoctorBob]

@virginialikesyou You're welcome! - Bob