Problem 23 - Thomas Calculus

Question

Find equations for the planes in Exercises 21–26. The plane through \((1,1,-1),(2,0,2),\) and \((0,-2,1)\)

Step-by-Step Solution

Answer

The plane equation is \(7x - 5y - 4z = 6\).

Step 1: Find Vectors in the Plane

To find the equation of the plane, we first need to find two vectors that lie in the plane defined by the given points. We can obtain these vectors by subtracting the coordinates of the given points. Let \( \vec{v_1} = (2, 0, 2) - (1, 1, -1) = (1, -1, 3) \) and \( \vec{v_2} = (0, -2, 1) - (1, 1, -1) = (-1, -3, 2) \).

Step 2: Find the Cross Product

To find a vector that is normal (perpendicular) to the plane, we calculate the cross product of \( \vec{v_1} \) and \( \vec{v_2} \). The cross product \( \vec{n} = \vec{v_1} \times \vec{v_2} \) is calculated as follows:\[\vec{n} = (1, -1, 3) \times (-1, -3, 2) \]\[= ((-1)(2) - (3)(-3), (3)(-1) - (1)(2), (1)(-3) - (-1)(-1))\]\[= (-2 + 9, -3 - 2, -3 - 1) = (7, -5, -4) \].Hence, the normal vector to the plane is \( (7, -5, -4) \).

Step 3: Write the Equation of the Plane

Using the normal vector \( (7, -5, -4) \) and a point on the plane, let's choose \((1, 1, -1)\), we can write the plane's equation in the form \( ax + by + cz = d \). Substituting, we have:\[7(x - 1) - 5(y - 1) - 4(z + 1) = 0\].Simplifying gives:\[7x - 5y - 4z = 7 - 5 + 4 = 6\].Thus, the equation of the plane is \(7x - 5y - 4z = 6\).

Key Concepts

Vectors in 3D Space

Understanding vectors in 3D space is crucial when dealing with spatial problems. In three dimensions, a vector is represented by an ordered triplet, say \(a, b, c\). This can be thought of as an arrow pointing from the origin to the point \(a, b, c\) in a 3D coordinate system.

Vectors help in describing movement and direction across space. Three key attributes define a vector:

• Magnitude: The length of the vector, calculated using the formula \(\sqrt{a^2 + b^2 + c^2}\).
• Direction: Indicated by the components \(a, b, c\).
• Position: While vectors themselves don't have a fixed location, they define movement or force in a specific direction from any start point.

Once you understand vectors' concepts, using them to define lines and planes becomes straightforward.

Cross Product

The cross product is a mathematical operation that can be performed on two vectors in three-dimensional space. This operation is denoted by the symbol \( \times \) and results in a third vector that is perpendicular to the original two.

For two vectors \(\vec{a} = (a_1, a_2, a_3)\) and \(\vec{b} = (b_1, b_2, b_3)\), the cross product \(\vec{a} \times \vec{b}\) is:

• \( \vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1) \)

This resulting vector is significant because:

• It is perpendicular to both \(\vec{a}\) and \(\vec{b}\).
• Its magnitude represents the area of the parallelogram spanned by the two vectors.

Using the cross product is essential in finding the normal vector to a plane, which helps in deriving the plane equation.

Normal Vector

A normal vector is pivotal when determining the characteristics of a plane in three-dimensional geometry. This vector is orthogonal, or perpendicular, to each vector lying within the plane, thus defining the plane's orientation in space.

To find a normal vector for a plane defined by two vectors, \(\vec{v_1}\) and \(\vec{v_2}\), you compute their "cross product." The normal vector is critical for the following reasons:

• It is used in plane equations of the form \(ax + by + cz = d\), where \(a\), \(b\), and \(c\) are the components of the normal vector.
• With an understanding of the normal vector, you can easily interpret the plane's tilt and positioning.

Remember, the normal vector does more than just define the plane's direction; it holds information about how the plane sits relative to other structures in 3D space.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometric figures using a coordinate system. It combines algebra and geometry to describe mathematical properties of figures using equations.

One of the main applications of coordinate geometry is finding the equations of geometric figures, such as lines and planes.
In three dimensions, any plane can be defined by a point and a normal vector. The standard equation is \(ax + by + cz = d\). Here's how it works:

• \(a\), \(b\), and \(c\) are the components of the normal vector.
• \(d\) is determined using a known point on the plane, substituting the coordinates into the equation to solve for \(d\).

This effective integration of geometry and algebra using coordinates facilitates solving complex spatial problems and understanding 3D relationships via mathematical expressions.