Video: Finding the Measure of the Angle between Two Planes

Find, to the nearest degree, the measure of the angle between the planes 2(𝑥 − 1) + 3(𝑦 − 4) + 4(𝑧 + 5) = 0 and 𝐫 ⋅ <1, −2, 5> = 16.

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Video Transcript

Find, to the nearest degree, the measure of the angle between the planes two times the quantity 𝑥 minus one plus three times the quantity 𝑦 minus four plus four times the quantity 𝑧 plus five equals zero and 𝐫 dot one negative two five equals 16.

Okay, so here we have these two planes given in equations of different forms. The equation that involves this 𝐫 vector, we’ll call this plane number two, is given in vector form, while the other plane, we’ll call it plane one, is nearly given in what’s called general form. As we look to solve for the angle between these two planes, we’ll start by finding a vector that’s normal to plane one and a vector that’s normal to plane two. We do this because the cosine of the angle between two planes is given by this expression, where 𝐧 one and 𝐧 two are vectors normal to those two planes.

Starting with our first plane’s equation, if we multiply through all the parentheses and then collect all the values, not multiplying a variable, we find that two 𝑥 plus three 𝑦 plus four 𝑧 plus six equals zero. Now, our plane is given in what’s called general form. And, in this form, the components of a vector normal to this plane are given by the values that multiply 𝑥, 𝑦, and 𝑧. If we call this vector that’s normal to our plane 𝐧 one, this will mean it has components two, three, and four.

Moving on to our second plane, as we saw, this is given in what’s called vector form. Written this way, we have a vector 𝐫 that goes to an arbitrary point in the plane, dotted with the vector that is normal to it. This means that right away we can read off the components of a vector normal to this plane, we’ll call it 𝐧 two. Now that we know the components of vectors normal to both of our planes, we can substitute them into this expression. The cosine of the angle between our two planes, plane one and plane two, is equal to the magnitude of the dot product of these two vectors divided by the product of the magnitude of each one.

If we begin to evaluate this dot product in our numerator and square out the various components in our denominator, we get this expression, which simplifies to 16 divided by the square root of 30 times the square root of 29. We recall that this is equal to the cos of the angle between our planes. And this means that 𝜃 itself equals the inverse cos of 16 over root 30 times root 29. Entering this expression in our calculator, to the nearest degree, it equals 57 degrees. This is the measure of the angle between our two planes.

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