Find the acute angle (in degrees) between the planes -2y + z = -2 and -3x + 3y = -5.

To find the acute angle between two planes, we can use the formula for the angle between two planes in terms of the dot product of their normal vectors.

The angle \(\theta\) between two planes with normal vectors \(\mathbf{n}_1\) and \(\mathbf{n}_2\) is given by:

\[
\cos \theta = \frac{|\mathbf{n}_1 \cdot \mathbf{n}_2|}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|}
\]

### Step 1: Identify the normal vectors of the planes

The given equations of the planes are:

1. \(-2y + z = -2\)
2. \(-3x + 3y = -5\)

For each plane, the coefficients of \(x\), \(y\), and \(z\) give us the components of the normal vector:

- For plane 1: \(\mathbf{n}_1 = (0, -2, 1)\)
- For plane 2: \(\mathbf{n}_2 = (-3, 3, 0)\)

### Step 2: Compute the dot product of the normal vectors

\[
\mathbf{n}_1 \cdot \mathbf{n}_2 = (0)(-3) + (-2)(3) + (1)(0) = 0 - 6 + 0 = -6
\]

### Step 3: Compute the magnitudes of the normal vectors

\[
\|\mathbf{n}_1\| = \sqrt{0^2 + (-2)^2 + 1^2} = \sqrt{0 + 4 + 1} = \sqrt{5}
\]

\[
\|\mathbf{n}_2\| = \sqrt{(-3)^2 + 3^2 + 0^2} = \sqrt{9 + 9 + 0} = \sqrt{18} = 3\sqrt{2}
\]

### Step 4: Compute the cosine of the angle

\[
\cos \theta = \frac{|\mathbf{n}_1 \cdot \mathbf{n}_2|}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|} = \frac{| -6 |}{\sqrt{5} \cdot 3\sqrt{2}} = \frac{6}{3\sqrt{10}} = \frac{2}{\sqrt{10}}
\]

### Step 5: Find the acute angle in degrees

Now we find the angle \(\theta\) using the inverse cosine function:

\[
\theta = \cos^{-1}\left(\frac{2}{\sqrt{10}}\right)
\]

Let's compute the angle in degrees.The acute angle between the planes \(-2y + z = -2\) and \(-3x + 3y = -5\) is approximately **50.77 degrees**.

Would you like further details on any of the steps, or do you have any additional questions?

Here are 5 related questions you might be interested in:
1. How do you find the angle between two lines in 3D space?
2. What is the geometric meaning of the dot product of two vectors?
3. How do you find the angle between two planes if they are given in parametric form?
4. How can you find the angle between a line and a plane?
5. What is the relation between the cross product and the sine of the angle between two vectors?

**Tip:** When finding the angle between planes, ensure that the normal vectors are correctly identified from the coefficients of the plane equations.

Explore how to find the acute angle between the planes -2y + z = -2 and -3x + 3y = -5 using vector concepts like dot product and magnitudes. This step-by-step guide includes key formulas and detailed calculations, ideal for Grades 11-12.

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