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Chapter 3: Problem 49

The intervepts of the plane \(2 x-3 y+z=12\) on the coordinate axes are

Short Answer

Expert verified
The intercepts are (6, 0, 0), (0, -4, 0), and (0, 0, 12).

Step by step solution

01

Understanding Plane Equation

The equation of the plane given is \(2x - 3y + z = 12\). This equation represents a plane in three-dimensional space. We will find the intercepts on the x-axis, y-axis, and z-axis.
02

Finding the x-intercept

To find the x-intercept, we set \(y = 0\) and \(z = 0\) and solve for \(x\). Substituting these values into the plane equation gives us: \(2x = 12\). Solving for \(x\), we find \(x = 6\). Thus, the x-intercept is \((6, 0, 0)\).
03

Finding the y-intercept

To find the y-intercept, set \(x = 0\) and \(z = 0\) in the plane equation. This results in \(-3y = 12\). Solving for \(y\), we get \(y = -4\). Hence, the y-intercept is \((0, -4, 0)\).
04

Finding the z-intercept

For the z-intercept, set \(x = 0\) and \(y = 0\). Substituting these into the plane equation gives \(z = 12\). Therefore, the z-intercept is \((0, 0, 12)\).
05

Conclusion on Intercepts

The intercepts of the plane \(2x - 3y + z = 12\) are thus \((6, 0, 0)\) on the x-axis, \((0, -4, 0)\) on the y-axis, and \((0, 0, 12)\) on the z-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Intercepts
Understanding intercepts is key to analyzing a plane in three-dimensional geometry. An intercept is a point where a line or surface crosses an axis. In the context of a plane defined by the equation \[2x - 3y + z = 12\]intercepts occur on the x, y, and z-axes.
  • **The X-Intercept**: This occurs where the plane intersects the x-axis. To find it, set both \(y\) and \(z\) to zero, then solve for \(x\). For our equation: \[2x = 12\]Solving gives \(x = 6\), making the intercept \((6, 0, 0)\).
  • **The Y-Intercept**: This is found where the plane meets the y-axis. Set \(x\) and \(z\) to zero, and solve for \(y\). The calculation for our plane is: \[-3y = 12\]The solution is \(y = -4\), making the intercept \((0, -4, 0)\).
  • **The Z-Intercept**: To find this, set \(x\) and \(y\) to zero. For the plane equation: \[z = 12\]The intercept is then \((0, 0, 12)\).
These intercepts provide tangible points, aiding in visualizing the position and orientation of the plane.
Coordinate Axes
In three-dimensional geometry, coordinate axes provide a reference framework that helps us understand the position of points in space. There are three perpendicular axes: the x-axis, the y-axis, and the z-axis.
  • **The X-Axis**: Represents the horizontal dimension. Points along this axis have the format \((x, 0, 0)\).
  • **The Y-Axis**: This is vertical in the plane parallel to the z-axis, typically formatted as \((0, y, 0)\).
  • **The Z-Axis**: Adds depth to our spatial understanding, formatted as \((0, 0, z)\).
Understand these axes helps in visualizing geometrical shapes.When analyzing the intercepts of a plane, we rely heavily on the concept of setting the other two coordinates to zero to find where the plane hits each axis.Coordination with these axes is crucial for determining intercepts, thus supporting the visualization and analytical understanding of three-dimensional objects.
Plane Equation
A plane equation in three-dimensional space is crucial to understanding spatial geometry. The standard form is \[ax + by + cz = d\]where \(a\), \(b\), and \(c\) are coefficients that dictate the orientation of the plane, and \(d\) shifts the plane's position.The plane equation \[2x - 3y + z = 12\]demonstrates how each coefficient affects the plane's interaction with each axis:
  • **Coefficient Impact**: - \(a = 2\), tends the plane more along the x-axis. - \(b = -3\), aligns it against the y-axis. - \(c = 1\), centers it with respect to the z-axis.The positive or negative signs further impact how the plane is tilted in space.
  • **The Constant \(d\)**: - The 12 pushes the plane away from the origin, dictating where the plane is positioned in relation to the coordination axes.
Understanding these aspects helps us to calculate intercepts and gain a more complex picture of spatial relationships in three-dimensional geometry.

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Most popular questions from this chapter

The equation of the plane pasning through \((4,-2,1)\) and perpendicular to the line with direction nation \(7,2,-3\) in (a) \(x+3 y-4 z-8=0\) (b) \(2 x+7 y-3 z-24=0\) \(\begin{array}{llll}\text { (c) } 7 x+2 y-3 z-21=0 . & \text { (d) } 7 x+3 y-2 z+21 & =0 . & \text { (V.T.t., } 2009 & S\end{array}\)

The line \(x=a y+b, z=c y+d\) and \(x=a^{\prime} y+b^{\prime}, z=c^{\prime} y+d^{\prime}\) are perpendicular if (a) \(a a^{\circ}+c c^{\prime}=1\) (b) \(a a^{\circ}+c c^{\prime}=-1\) (c) \(b b^{2}+d d^{\prime}=1\) (d) \(b b^{\prime}+d d^{\prime}=-1\).

The line \(\frac{x-1}{1}=\frac{y-2}{2}=\frac{x-3}{3}\) and the plane \(x+y-z=0\) are (a) parallel (b) perpendicular (e) such that the line lies in the plane.

The equation of a right eircular cylinder, whose axis in the \(z\)-axis and radius \(a\) is \((a) x^{2}+y^{2}+z^{2}=a^{2}\) (b) \(a^{2}+y^{2}=a^{2}\) (c) \(x^{2}+y^{2}=a^{2}\) \(\left(\right.\) d) \(2^{2}+x^{2}=a^{2}\)

A line malies angles \(o, \beta, \gamma\) with the coordinate axes, then ces \(2 \alpha+\cos 2 \beta+\cos 2 \gamma\) is equal to (a) 1 (b) 2 (c) \(-1\) \((d)-2\)
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