Question

Sketch a graph of the line and find the points (if any) where the line intersects the \(x y-, x z-,\) and \(y z\) -planes. $$ \frac{x-2}{3}=y+1=\frac{z-3}{2} $$

Short answer

The points of intersection with the xy-plane, xz-plane and yz-plane are (-2.5, -2.5, 0), (2, 0, 3) and (0, 0.5, 3) respectively. The corresponding graph can be plotted by drawing a line connecting these points in 3D space.

Step-by-step solution

Identify the points where the line intersects the xy, xz, and yz planes

To find the intersection point with the xy-plane, make \(z=0\) and solve for \(x\) and \(y\). Similarly, to find the intersection point with the xz-plane, make \(y=0\) and solve for \(x\) and \(z\). Lastly, to find the intersection with the yz-plane, make \(x=0\) and solve for \(y\) and \(z\).

Calculation of intersection points

For intersection with xy-plane (\(z = 0\)), using the provided equation \(\frac{x-2}{3}=y+1=\frac{z-3}{2}\), upon substituting \(z=0\), we get: \(\frac{x-2}{3}=y+1=-\frac{3}{2}\). Solving these equations give \(x = -\frac{3}{2} * 3 + 2 = -2.5\) and \(y = -\frac{3}{2} - 1 = -2.5\). Thus intersection point with xy-plane is (-2.5, -2.5, 0). For intersection with xz-plane (\(y = 0\)), substituting \(y=0\) in the equation, we get: \(\frac{x-2}{3}=0=\frac{z-3}{2}\). Solving these equations give \(x = 2\) and \(z = 3\). Therefore, intersection point with xz-plane is (2, 0, 3). For intersection with yz-plane (\(x = 0\)), substituting \(x=0\) in the equation, we get: \(y+1=\frac{z-3}{2}\). Solving for these variables gives: \(y = \frac{3}{2} - 1 = 0.5\) and \(z = 3\). Thus, the intersection point with yz-plane is (0, 0.5, 3).

Sketch the graph from intersection points

The next step is to sketch the graph. Plot the intersection points -2.5 on the x-axis, -2.5 on the y-axis, and 0 on the z-axis for the point where the line crosses the xy-plane. Similarly, plot 2 on the x-axis, 0 on the y-axis, and 3 on the z-axis for the point where the line crosses the xz-plane. Plot 0 on the x-axis, 0.5 on the y-axis, and 3 on the z-axis for the point where the line crosses the yz-plane. Then, connect those points to form the line in the 3D space.