To determine the equation of a plane when you have its x, y, and z intercepts, you can use the intercept form of the equation of a plane. The intercepts are as follows:
- X-intercept (a): This is the point where the plane intersects the x-axis. In this case, it is 2, which means the plane intersects the x-axis at (2, 0, 0).
- Y-intercept (b): This is the point where the plane intersects the y-axis. Here, it is 3, indicating the intersection at (0, 3, 0).
- Z-intercept (c): This is the point where the plane intersects the z-axis, which is 1 in this scenario. The intersection occurs at (0, 0, 1).
Knowing these intercepts, you can write the equation of the plane in the following intercept form:
rac{x}{a} + rac{y}{b} + rac{z}{c} = 1
Substituting the values of the intercepts:
rac{x}{2} + rac{y}{3} + rac{z}{1} = 1
This equation can be simplified for clarity:
Multiplying through by 6 (the least common multiple of the denominators 2, 3, and 1) to eliminate the fractions:
6*( rac{x}{2}) + 6*( rac{y}{3}) + 6*( rac{z}{1}) = 6*
Which simplifies to:
3x + 2y + 6z = 6
Thus, the equation of the plane with the given intercepts is: