Stanton D. answered • 10/26/14

Tutor to Pique Your Sciences Interest

Dalia,

Think of the LEFT side of the 3x+2y-5z=-4 equation as setting up a relationship among the *orientation* of that plane with respect to the various axes (x, y, and z) and the RIGHT side (-4) as indicating the perpendicular *distance* of that plane from the origin (0,0,0) point. (In fact, that's just what the equation says!)

So, what would be true about any plane parallel to that plane? It would have the same *orientation* setup (3x+2y-5z) but it would be a *different* distance perpendicularly from the origin (=??, something other than -4).

So, all you have to do, is plug the given point *into* that equation, and find out what the ?? calculates to:

3x+2y-5z=3(-3)+2(-1)-5(-2) = -9-2+10 = -1.

SO, the new plane, the one that contains the desired point, has the equation:

3x+2y-5z=-1.

In fact, this plane is 3 units closer to the origin (0,0,0) than the original plane was, but parallel to it, just as you wanted.

Incidentally, have you thought about what

**physical significance**those coefficients 3, 2, and -5 have? You should probably do that, maybe starting with the simpler case of a line equation in the x-y plane. When you've done that, and compared the equation to its graph for a few cases, you'll make the "discovery" that those coefficient numbers describe ***inversely*** how close the line is to the origin for (i.e. in the axis direction of) each of those variables. For instance, a line with the equation:3x + 5y = 6

has an x-intercept (because y=0 there) of 6/3 = 2

and a y-intercept (because x=0 there) of 6/5 = 1.2

You might say, the line is "close to the origin" along the y axis by 1.2 units distance but

a little "further from the origin" along the x-axis (namely, 2 units distance).

The take-home message here is that a BIG number as coefficient for a variable means the item (a line, a plane, etc.) is CLOSE in that variable direction, but that a SMALL number as coefficient for a variable means the same item is FURTHER AWAY in that variable direction. Similarly, a POSITIVE coefficient for a variable (and a positive constant on the other side of the equation) means the item lies towards (that is, has its intercept on that variable axis) in the POSITIVE direction of the axis, whereas a NEGATIVE coefficient for a variable (with a positive constant on the other side of the equation) means the item lies towards the NEGATIVE direction of the axis.

In the above case, since the constant term is negative (less than zero) for both planes, the directions towards the intercepts is reversed: for the original plane, the intercepts with the x, y, and z axes respectively were

x: -4/3 = -4/3 (that's from 3x+2(0)-5(0)=-4, solve for x) and so on:

y: -4/2 = -2

z: -4/-5 = 4/5

Definitely work through some equations and their graphed lines as I suggest above, and suddenly you'll really

**understand**what's going on with equations and graphs. It's a great feeling when you do**get**it!