Problem 2
Question
Find the intercepts and sketch the graph of the plane. $$ 3 x+6 y+2 z=6 $$
Step-by-Step Solution
Verified Answer
The intercepts of the plane equation 3x + 6y + 2z = 6 are: x-intercept is 2, y-intercept is 1, and z-intercept is 3. The plane can be graphed by plotting these intercepts in three-dimensional space and connecting them to form the plane.
1Step 1: Find the x-intercept
To find the x-intercept, set y = 0 and z = 0 in the equation. This gives 3x + 6(0) + 2(0) = 6, which simplifies to 3x = 6. Solving for x, we get x = 2.
2Step 2: Find the y-intercept
Similarly, to find the y-intercept, set x = 0 and z = 0 in the equation. This gives 3(0) + 6y + 2(0) = 6, which simplifies to 6y = 6. Solving for y, we get y = 1.
3Step 3: Find the z-intercept
Similarly, to find the z-intercept, set x = 0 and y = 0 in the equation. This gives 3(0) + 6(0) + 2z = 6, which simplifies to 2z = 6. Solving for z, we get z = 3.
4Step 4: Sketch the plane
Plot the x, y, and z intercepts on a three-dimensional graph. The x-intercept is (2,0,0), the y-intercept is (0,1,0), and the z-intercept is (0,0,3). Draw a triangle connecting the three intercepts. These three lines form the edges of the plane. Finally, extend the triangle to form the plane.
Key Concepts
X-Intercept CalculationY-Intercept CalculationZ-Intercept Calculation3D Graph Sketching
X-Intercept Calculation
When attempting to find the x-intercept of a plane in 3D space, represented by the equation 3x + 6y + 2z = 6, we look for the point where the plane crosses the x-axis. This is found by setting the values of the other variables, y and z, to zero.
By substituting y = 0 and z = 0 into our equation, it simplifies to 3x = 6. We then solve for x, which represents the distance from the origin to the point where the plane intersects the x-axis. In this particular equation, the x-intercept is calculated to be x = 2, hence the point of intersection is (2,0,0).
By substituting y = 0 and z = 0 into our equation, it simplifies to 3x = 6. We then solve for x, which represents the distance from the origin to the point where the plane intersects the x-axis. In this particular equation, the x-intercept is calculated to be x = 2, hence the point of intersection is (2,0,0).
Y-Intercept Calculation
Locating the y-intercept requires us to find where the plane meets the y-axis. This means both x and z must be zero.
Using the equation 3x + 6y + 2z = 6 and setting x = 0 and z = 0, we're left with 6y = 6. Solving for y gives us the value of 1, which represents the y-intercept of the plane. Subsequently, the coordinate of the y-intercept is noted as (0,1,0).
Using the equation 3x + 6y + 2z = 6 and setting x = 0 and z = 0, we're left with 6y = 6. Solving for y gives us the value of 1, which represents the y-intercept of the plane. Subsequently, the coordinate of the y-intercept is noted as (0,1,0).
- Set x and z to zero to focus on the y-variable.
- Solve the resulting equation for y.
- Plot this point on the y-axis to represent the y-intercept.
Z-Intercept Calculation
To determine where the plane cuts through the z-axis, known as the z-intercept, we set both x and y to zero in the original equation. This method isolates the z variable.
In the given equation, introducing x = 0 and y = 0 turns the equation into 2z = 6. Solving this, z is found to be 3, therefore the z-intercept is at (0,0,3).
In the given equation, introducing x = 0 and y = 0 turns the equation into 2z = 6. Solving this, z is found to be 3, therefore the z-intercept is at (0,0,3).
- Isolate z by making x and y zero.
- Solve for z to get the z-intercept.
- The z-intercept point is then plotted on the z-axis.
3D Graph Sketching
Sketching a plane within a 3D coordinate system entails plotting the x, y, and z-intercepts and connecting them to form a triangular shape, which acts as part of the plane.
For our equation 3x + 6y + 2z = 6, we have the intercepts (2,0,0), (0,1,0), and (0,0,3). We start by plotting these points on their respective axes. After marking these points, we draw lines to connect them, forming a triangle that is a portion of the plane. This shape helps visualize the orientation and position of the plane. Finally, to represent the actual plane, we extend this triangle infinitely along its breadth and width, making sure that it remains consistent with its intercepts.
Through practice, sketching a 3D graph can become intuitive, aiding in better comprehension of the spatial relationship between planes and the coordinate axes.
For our equation 3x + 6y + 2z = 6, we have the intercepts (2,0,0), (0,1,0), and (0,0,3). We start by plotting these points on their respective axes. After marking these points, we draw lines to connect them, forming a triangle that is a portion of the plane. This shape helps visualize the orientation and position of the plane. Finally, to represent the actual plane, we extend this triangle infinitely along its breadth and width, making sure that it remains consistent with its intercepts.
Through practice, sketching a 3D graph can become intuitive, aiding in better comprehension of the spatial relationship between planes and the coordinate axes.
Other exercises in this chapter
Problem 2
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