Find the acute angle (in degrees) between the planes x+4y-5z=-6 and 4x-4y-5z=7.

To find the acute angle between two planes, we use the formula:

\[
\cos \theta = \frac{| \vec{n_1} \cdot \vec{n_2} |}{|\vec{n_1}| |\vec{n_2}|}
\]

where:
- \( \theta \) is the angle between the two planes.
- \( \vec{n_1} = \langle A_1, B_1, C_1 \rangle \) is the normal vector of the first plane \( A_1x + B_1y + C_1z = D_1 \).
- \( \vec{n_2} = \langle A_2, B_2, C_2 \rangle \) is the normal vector of the second plane \( A_2x + B_2y + C_2z = D_2 \).

### Step 1: Find the normal vectors of the planes.
The normal vector of the plane \( x + 4y - 5z = -6 \) is:
\[
\vec{n_1} = \langle 1, 4, -5 \rangle
\]

The normal vector of the plane \( 4x - 4y - 5z = 7 \) is:
\[
\vec{n_2} = \langle 4, -4, -5 \rangle
\]

### Step 2: Compute the dot product \( \vec{n_1} \cdot \vec{n_2} \).
\[
\vec{n_1} \cdot \vec{n_2} = (1)(4) + (4)(-4) + (-5)(-5) = 4 - 16 + 25 = 13
\]

### Step 3: Compute the magnitudes \( |\vec{n_1}| \) and \( |\vec{n_2}| \).
\[
|\vec{n_1}| = \sqrt{1^2 + 4^2 + (-5)^2} = \sqrt{1 + 16 + 25} = \sqrt{42}
\]
\[
|\vec{n_2}| = \sqrt{4^2 + (-4)^2 + (-5)^2} = \sqrt{16 + 16 + 25} = \sqrt{57}
\]

### Step 4: Compute \( \cos \theta \).
\[
\cos \theta = \frac{|13|}{\sqrt{42} \times \sqrt{57}} = \frac{13}{\sqrt{2394}}
\]

### Step 5: Find \( \theta \) by taking the inverse cosine.
\[
\theta = \cos^{-1} \left( \frac{13}{\sqrt{2394}} \right)
\]

Let's calculate the exact value.The acute angle between the two planes is approximately \( 74.59^\circ \).

Would you like further details or have any questions? Here are 5 related questions to explore:

1. How do we calculate the angle between two vectors?
2. What is the significance of the normal vector in geometry?
3. How would the process change if one of the planes was parallel to an axis?
4. Can the angle between planes be obtuse? How would that be reflected in the cosine calculation?
5. How are normal vectors used in physics, especially in relation to forces?

**Tip:** The dot product is an important operation in vector mathematics and helps find angles and projections between vectors.

Find the acute angle (in degrees) between the planes x+4y-5z=-6 and 4x-4y-5z=7.

Learn how to find the acute angle between two planes, x+4y-5z=-6 and 4x-4y-5z=7, using vector dot product and trigonometric formulas. This detailed explanation covers geometry and vector mathematics, suitable for students in Grades 10-12.

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