To find the acute angle between the two planes $3x - y + 2z = 5$ and $x + 3y - 3z = 5$, we need to calculate the angle between their normal vectors.

Step 1: Identify the Normal Vectors

The general form of a plane equation is $Ax + By + Cz = D$, where $\vec{n} = \langle A, B, C \rangle$ is the normal vector.

For the planes given:

• Plane 1: $3x - y + 2z = 5$ has a normal vector $\vec{n_1} = \langle 3, -1, 2 \rangle$.
• Plane 2: $x + 3y - 3z = 5$ has a normal vector $\vec{n_2} = \langle 1, 3, -3 \rangle$.

Step 2: Use the Dot Product to Find the Angle

The angle $\theta$ between two vectors $\vec{n_1}$ and $\vec{n_2}$ can be found using the dot product formula: $\cos \theta = \frac{\vec{n_1} \cdot \vec{n_2}}{|\vec{n_1}| |\vec{n_2}|}$

Calculate the Dot Product $\vec{n_1} \cdot \vec{n_2}$

$\vec{n_1} \cdot \vec{n_2} = (3)(1) + (-1)(3) + (2)(-3) = 3 - 3 - 6 = -6$

Calculate the Magnitudes $|\vec{n_1}|$ and $|\vec{n_2}|$

$|\vec{n_1}| = \sqrt{3^2 + (-1)^2 + 2^2} = \sqrt{9 + 1 + 4} = \sqrt{14}$ $|\vec{n_2}| = \sqrt{1^2 + 3^2 + (-3)^2} = \sqrt{1 + 9 + 9} = \sqrt{19}$

Substitute into the Cosine Formula

$\cos \theta = \frac{-6}{\sqrt{14} \cdot \sqrt{19}}$

Calculate $\theta$

Using the inverse cosine function, we find: $\theta = \cos^{-1} \left( \frac{-6}{\sqrt{14} \cdot \sqrt{19}} \right)$

Using a calculator to approximate the value in radians (rounded to three significant figures): $\theta \approx 1.89 \text{ radians}$

So, the acute angle between the planes is approximately 1.89 radians.

Would you like further details on any part of this calculation?

Related Questions

1. How do you find the angle between two lines instead of planes?
2. What is the geometric interpretation of the dot product in finding angles?
3. How would you calculate angles between planes in 4-dimensional space?
4. Can the angle between planes be larger than 90 degrees?
5. How would the solution change if the planes were parallel?

Tip

When working with planes, always verify the direction of normal vectors as they influence the orientation and the resulting angle between the planes.