Problem 94 - Calculus Early Transcendentals

Question

Find the points at which the plane \(a x+b y+c z=d\) intersects the \(x-y-\), and \(z\) -axes.

Step-by-Step Solution

Answer

Answer: The points at which the plane ax + by + cz = d intersects the x-y, y-z, and z-axes are: 1. x-axis intersection: \((\frac{d}{a}, 0, 0)\) 2. y-axis intersection: \((0, \frac{d}{b}, 0)\) 3. z-axis intersection: \((0, 0, \frac{d}{c})\)

Step 1: Find intersection with the x-axis

To find the point where the plane intersects the x-axis, set the y and z coordinates equal to zero in the equation of the plane. The equation we are given is \(ax + by + cz = d\). Setting \(y=0\) and \(z=0\), we get: $$a x + b(0) + c(0) = d$$ Solving for \(x\), we get the intersection point with the x-axis: $$x = \frac{d}{a}$$ So, the intersection point with the x-axis is \((\frac{d}{a}, 0, 0)\).

Step 2: Find intersection with the y-axis

Similarly, to find the point where the plane intersects the y-axis, set the x and z coordinates equal to zero in the equation of the plane. Setting \(x=0\) and \(z=0\), we get: $$a(0) + b y + c(0) = d$$ Solving for \(y\), we get the intersection point with the y-axis: $$y = \frac{d}{b}$$ The intersection point with the y-axis is \((0, \frac{d}{b}, 0)\).

Step 3: Find intersection with the z-axis

Lastly, to find the point where the plane intersects the z-axis, set the x and y coordinates equal to zero in the equation of the plane. Setting \(x=0\) and \(y=0\), we get: $$a(0) + b(0) + c z = d$$ Solving for \(z\), we get the intersection point with the z-axis: $$z = \frac{d}{c}$$ The intersection point with the z-axis is \((0, 0, \frac{d}{c})\).

Step 4: Final Answer

The points at which the plane \(ax + by + cz = d\) intersects the x-y, y-z, and z-axes are: 1. x-axis intersection: \((\frac{d}{a}, 0, 0)\) 2. y-axis intersection: \((0, \frac{d}{b}, 0)\) 3. z-axis intersection: \((0, 0, \frac{d}{c})\)

Key Concepts

Understanding Coordinate Geometry

Coordinate geometry is a branch of mathematics that combines algebraic equations with geometric figures through a coordinate system. In 3D coordinate geometry, points, lines, and planes are described using three coordinates: x, y, and z. This system allows us to precisely define the position of points in a three-dimensional space, and to explore their relationships and intersections.

One powerful aspect of coordinate geometry is its ability to simplify finding intersections in space. By setting the coordinates of a plane equation to zero, we can find how it cuts through the different axes. This means we're calculating where a plane, represented as an algebraic equation, intersects the x, y, and z axes. These intersection points give us the locations where the plane touches these axis lines. By mastering this, students can analyze more complex geometric problems with ease.

Exploring 3D Geometry

3D geometry extends the principles of traditional geometry into three-dimensional space, offering a whole new realm of exploration. In a 3D system, each point has three coordinates, typically written as \(x, y, z\). This allows us to describe points precisely in three dimensions.

Understanding 3D geometry is crucial in examining the relationships and properties of different geometric figures. For planes in particular, we consider how they are situated in 3D space and how they interact with other geometric entities. Planes can intersect lines, other planes, or be positioned parallel to coordinate axes. Recognizing these interactions is key to solving problems involving spatial reasoning. This helps build a foundational understanding that is crucial for fields like physics, engineering, and computer graphics.

Equations of Planes in 3D Geometry

The equation of a plane in its simplest form is written as \(ax + by + cz = d\). This represents all the points \(x, y, z\) that lie on the plane in three-dimensional space.

• \( a, b, c \) are the coefficients that determine the orientation of the plane.
• \( d \) is the constant that shifts the plane up or down along the normal vector.

To find where a plane intersects the x, y, or z-axes, we set two coordinates to zero and solve for the third. This is because at any given axis, two of the coordinates are zero by definition:

• On the x-axis, the y and z coordinates are zero.
• On the y-axis, the x and z coordinates are zero.
• On the z-axis, the x and y coordinates are zero.

By fully grasping how the plane's equation is structured and used, students can predict and verify intersection points quickly and confidently. This understanding is not only pivotal for solving exercises like this but also enhances spatial visualization skills which are vital in practical applications.