Problem 50 - Thomas Calculus

Question

Use a calculator to find the acute angles between the planes in Exercises \(49-52\) to the nearest hundredth of a radian. $$ x+y+z=1, \quad z=0 \quad \text (the\quad xy-plane) $$

Step-by-Step Solution

Answer

The acute angle between the planes is approximately 0.955 radians.

Step 1: Identify the Normal Vectors

The normal vector to the plane given by the equation \(x+y+z=1\) is \(\mathbf{n}_1 = \langle 1, 1, 1 \rangle\). The xy-plane, represented by \(z=0\), has a normal vector \(\mathbf{n}_2 = \langle 0, 0, 1 \rangle\).

Step 2: Use Dot Product to Find the Angle

The angle \(\theta\) between the two planes' normal vectors is given by the formula \(\cos \theta = \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|}\).

Step 3: Calculate the Dot Product

Compute the dot product \(\mathbf{n}_1 \cdot \mathbf{n}_2 = \langle 1, 1, 1 \rangle \cdot \langle 0, 0, 1 \rangle = 1 \times 0 + 1 \times 0 + 1 \times 1 = 1\).

Step 4: Calculate the Magnitudes of the Vectors

Find \(\|\mathbf{n}_1\| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}\) and \(\|\mathbf{n}_2\| = \sqrt{0^2 + 0^2 + 1^2} = 1\).

Step 5: Solve for the Cosine of the Angle

Substitute these values into the equation: \(\cos \theta = \frac{1}{\sqrt{3}\times 1} = \frac{1}{\sqrt{3}}\).

Step 6: Calculate the Angle in Radians

Use a calculator to find \(\theta = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right) \approx 0.955\) radians.

Key Concepts

Normal Vectors

A normal vector is a vector that is perpendicular to a surface or a plane. It helps us understand the orientation of the plane in three-dimensional space. Consider a plane defined by an equation like \(x + y + z = 1\). The coefficients of \(x, y, \text{and } z\) (in this case, 1, 1, and 1) become the components of the normal vector \(\mathbf{n}_1 = \langle 1, 1, 1 \rangle\).

For the xy-plane, described by \(z=0\), the normal vector \(\mathbf{n}_2\) is \(\langle 0, 0, 1 \rangle\). This vector points directly along the z-axis, indicating that the xy-plane is flat and spans the x and y directions entirely.

Normal vectors are crucial because they simplify the process of finding the angle between planes and are used extensively in 3D modeling, physics simulations, and engineering designs. By using normal vectors, we can determine properties such as the distance between planes or whether they are parallel or perpendicular.

Dot Product

The dot product, also called the scalar product, is a way to multiply two vectors that results in a scalar (a single number). This value gives us insight into the relationship between the direction of the vectors. It is crucial in calculating angles and determining orthogonal properties of vectors.

For two vectors \(\mathbf{a} = \langle a_1, a_2, a_3 \rangle\) and \(\mathbf{b} = \langle b_1, b_2, b_3 \rangle\), their dot product is given by:

• \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\)

In our exercise, the dot product of the normal vectors \(\mathbf{n}_1 = \langle 1, 1, 1 \rangle\) and \(\mathbf{n}_2 = \langle 0, 0, 1 \rangle\) is:\

• \(\mathbf{n}_1 \cdot \mathbf{n}_2 = 1 \times 0 + 1 \times 0 + 1 \times 1 = 1\)

The dot product being non-zero confirms that the vectors are not perpendicular. This dot product value is then used in calculating the cosine of the angle between the planes.

Planes Angle Calculation

To find the angle between two planes, we first need their normal vectors. The relationship between these vectors is key, as the angle between the planes is the same as the angle between their normal vectors.

To find this angle \(\theta\), we use:

• \(\cos \theta = \frac{\mathbf{n}_1 \cdot \mathbf{n}_2}{\|\mathbf{n}_1\| \|\mathbf{n}_2\|}\)

Here, \(\|\mathbf{n}_1\|\) and \(\|\mathbf{n}_2\|\) are the magnitudes of the normal vectors. In our example:

• \(\|\mathbf{n}_1\| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}\)
• \(\|\mathbf{n}_2\| = \sqrt{0^2 + 0^2 + 1^2} = 1\)

Substitute these into the cosine formula to find \(\cos \theta = \frac{1}{\sqrt{3}}\). The final angle \(\theta\) is calculated using the arccosine function on a calculator, yielding approximately 0.955 radians.

This angle between the planes helps us understand their spatial relationship and is used in fields like robotics and computer graphics to control angles and rotations.