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Chapter 12: Problem 3

Where does the plane \(-2 x-3 y+4 z=12\) intersect the coordinate axes?

Short Answer

Expert verified
Answer: The points of intersection are `(-6, 0, 0)` for the x-axis, `(0, -4, 0)` for the y-axis, and `(0, 0, 3)` for the z-axis.

Step by step solution

01

Find the intersection with the x-axis (y=0, z=0)

To find the intersection with the x-axis, we need to set y and z equal to zero in the equation of the plane: $$-2x - 3(0) + 4(0) = 12.$$ Now, we can solve for x: $$-2x = 12 \Rightarrow x = -6.$$ So, the point of intersection with the x-axis is `(-6, 0, 0)`.
02

Find the intersection with the y-axis (x=0, z=0)

To find the intersection with the y-axis, we need to set x and z equal to zero in the equation of the plane: $$-2(0) - 3y + 4(0) = 12.$$ Now, we can solve for y: $$-3y = 12 \Rightarrow y = -4.$$ So, the point of intersection with the y-axis is `(0, -4, 0)`.
03

Find the intersection with the z-axis (x=0, y=0)

To find the intersection with the z-axis, we need to set x and y equal to zero in the equation of the plane: $$-2(0) - 3(0) + 4z = 12.$$ Now, we can solve for z: $$4z = 12 \Rightarrow z = 3.$$ So, the point of intersection with the z-axis is `(0, 0, 3)`.
04

Final Answer

The points where the plane intersects the coordinate axes are `(-6, 0, 0)` for the x-axis, `(0, -4, 0)` for the y-axis, and `(0, 0, 3)` for the z-axis.

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Most popular questions from this chapter

Flow in a cylinder Poiseuille's Law is a fundamental law of fluid dynamics that describes the flow velocity of a viscous incompressible fluid in a cylinder (it is used to model blood flow through veins and arteries). It says that in a cylinder of radius \(R\) and length \(L,\) the velocity of the fluid \(r \leq R\) units from the center-line of the cylinder is \(V=\frac{P}{4 L \nu}\left(R^{2}-r^{2}\right),\) where \(P\) is the difference in the pressure between the ends of the cylinder and \(\nu\) is the viscosity of the fluid (see figure). Assuming that \(P\) and \(\nu\) are constant, the velocity \(V\) along the center line of the cylinder \((r=0)\) is \(V=k R^{2} / L,\) where \(k\) is a constant that we will take to be \(k=1.\) a. Estimate the change in the centerline velocity \((r=0)\) if the radius of the flow cylinder increases from \(R=3 \mathrm{cm}\) to \(R=3.05 \mathrm{cm}\) and the length increases from \(L=50 \mathrm{cm}\) to \(L=50.5 \mathrm{cm}.\) b. Estimate the percent change in the centerline velocity if the radius of the flow cylinder \(R\) decreases by \(1 \%\) and the length \(L\) increases by \(2 \%.\) c. Complete the following sentence: If the radius of the cylinder increases by \(p \%,\) then the length of the cylinder must increase by approximately __________ \(\%\) in order for the velocity to remain constant.

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