Problem 42 - Calculus for Biology and Medicine

Question

Find the equation of the plane through \((3,-1,2)\) and perpendicular to \([-1,1,2]^{\prime} .\)

Step-by-Step Solution

Answer

The equation of the plane is \(-x + y + 2z = 0\).

Step 1: Identify Normal Vector and Known Point

The vector perpendicular to the plane, known as the normal vector, is \([-1, 1, 2]\). The known point through which the plane passes is \((3, -1, 2)\). These are the key pieces of information needed to find the plane equation.

Step 2: Use the Plane Equation Formula

The general equation of a plane with normal vector \(\vec{n} = [a, b, c]\) that passes through a point \((x_0, y_0, z_0)\) is given by:\[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \]

Step 3: Substitute Values into the Equation

Substitute the normal vector components \([-1, 1, 2]\) into \(a\), \(b\), \(c\), and the point \((3, -1, 2)\) into \((x_0, y_0, z_0)\):\[ -1(x - 3) + 1(y + 1) + 2(z - 2) = 0 \]

Step 4: Simplify the Equation

Distribute and simplify the equation:\[ -x + 3 + y + 1 + 2z - 4 = 0 \]\[ -x + y + 2z = 0 \]

Step 5: Rewrite the Simplified Plane Equation

Combine like terms and simplify further:\[ -x + y + 2z = 0 \] This is the equation of the plane.

Key Concepts

Normal Vector

In geometry, the normal vector is like a directional arrow that is perpendicular to a surface. It tells us which way is "out" or "away" from the surface. Think of it as a guide that ensures we know the exact orientation.
To picture this, imagine a flat table. The normal vector would be an arrow sticking straight up from the center of the table. Regardless of where you are on the table, this vector points away from the surface.
In our problem, the normal vector given is \([-1, 1, 2]\). This specific vector indicates the direction perpendicular to the plane, playing a crucial role in defining the plane's orientation. When dealing with planes in mathematics, knowing the normal vector is key to writing the equation of the plane.

Perpendicular

The concept of being perpendicular is essential when understanding planes and their normal vectors. Two things are perpendicular if they meet at a right angle, denoted by 90 degrees. When you think of the word "perpendicular," think of two lines or surfaces forming an 'L' shape.
In the context of planes, a normal vector is a vector that is perpendicular to every line that lies on the plane. This perpendicular relationship helps determine the alignment of the plane. Knowing that our normal vector is \([-1, 1, 2]\), we can deduce that it is standing straight out from the plane, providing a benchmark for how the plane is oriented in 3D space.

Plane Equation Formula

The plane equation formula is fundamental when it comes to geometry and algebra applications. It’s like a blueprint that shows us how to form a visual of a plane mathematically.
The formula is given by:

• \(a(x - x_0) + b(y - y_0) + c(z - z_0) = 0\)

In this formula, \(a, b,\) and \(c\) are the components of the normal vector, and \((x_0, y_0, z_0)\) is a known point on the plane. This known point helps anchor the equation to a specific location in space.
In the problem at hand, we used the normal vector \([-1, 1, 2]\) and the point \((3, -1, 2)\) to derive the plane equation. Simply plug these into the formula to find the specific plane that meets these conditions, resulting in an equation that describes the plane precisely through these coordinates.