Finding Plane Intercepts

To find the intercepts of the plane $3x - 4y - 2z = 6$ with the coordinate axes, we need to convert the equation into the intercept form: $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$. Here, $a$, $b$, and $c$ represent the x, y, and z intercepts, respectively.

Equation Conversion

First, divide the given equation $3x - 4y - 2z = 6$ by 6 to make the right-hand side equal to 1:

$ \frac{3x}{6} - \frac{4y}{6} - \frac{2z}{6} = \frac{6}{6} $

Simplify the fractions:

$ \frac{x}{2} - \frac{y}{\frac{6}{4}} - \frac{z}{\frac{6}{2}} = 1 $

$ \frac{x}{2} - \frac{y}{\frac{3}{2}} - \frac{z}{3} = 1 $

Identifying Intercepts

Rewrite the equation to match the standard intercept form $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$. We adjust the signs in the denominators:

$ \frac{x}{2} + \frac{y}{-\frac{3}{2}} + \frac{z}{-3} = 1 $

By comparing this with the standard form, we can identify the intercepts:

• x-intercept ($a$): 2
• y-intercept ($b$): $-\frac{3}{2}$
• z-intercept ($c$): -3

Conclusion

The intercepts made by the plane $3x - 4y - 2z = 6$ with the coordinate axes are 2, $-\frac{3}{2}$, and -3.