calculus

Question

calculus

davidusa · Answer

1. How do you derive the line and plane equations?
2. What are the line and plane formulas/equations?
3.When its best to use each formula/equations (if there are multiple like in r2- there is point slope formula and the slope intercept )
4. How are they useful and how do you use them in math?

mathmale · Answer

2. What are the line and plane formulas/equations?
There are several forms of the equation of a line in 3D space.
It's probably best to start with the vector form of such a line.
Suppose that you are given a point in space, $$(x _{0},y _{0},z _{0})$$
and are told that your line passes thru that point.  
Suppose that you are also told that the line has direction given by the vector

davidusa · Answer

ok. i understand that so far.

mathmale · Answer

$$v = .$$ Then the vector equation of that line in space is$$r=r _{0}+t$$ where both r and r-sub-0, as well as , represent VECTORS, and t is a scalar parameter.

davidusa · Answer

What is a scalar parameter?

mathmale · Answer

If we're told that a line passes through (1, -1, 4) and has direction vector v=<1,1,0>, then the vector equation of that line is what? $$r=< ? , ? , ? > + t < ? , ? , ? >$$

davidusa · Answer

r=+t<1,1,0>

mathmale · Answer

If v is the direction vector, and we multiply that vector v by a scalar called t, then t*v ("t times the vector v") is a new vector with exactly the same direction as the original vector v, but with greater or lesser length.  Again, we call t a "scalar parameter"; it is a multiplier of vector v.

davidusa · Answer

ok i understand that.

mathmale · Answer

Re: good start. r=+t<1,1,0> is the vector from the origin, (0,0,0) to the given point (1,-1,4). This info comes from my previous statement, "If we're told that a line passes through (1, -1, 4)". So, try again. Write out the vector equation, for this situation.

davidusa · Answer

oh ok. so r=<1,-1,4>+t<1,1,0> right?

mathmale · Answer

Great.  Now you're on track.  I felt it important to work on a SPECIFIC QUESTION such as this one, to ensure that you understood the definition and concepts involved and could apply them.

mathmale · Answer

At the risk of being repetitive, remember that that " r " should be BOLD or have an arrow over it, like this:|dw:1396027707042:dw|