# Lines and planes

1. Apr 28, 2012

### ronho1234

(a) Find the equation of the plane p which passes through the three points
(A 1,0,1), B(2,−1,1) .and C(0,3,2) .

(b) Find a scalar parametric form of the equation for the line which passes
through the point D(−1,1,1) and which is perpendicular to the plane p.

(c) Let E be the point where the line  intersects the plane p. Find, in the
scalar parametric equation for the line, the value of the parameter which
corresponds to the point E and hence find the co-ordinates of this point.

(d) What is the closest distance of the point D from the plane p?

i've done most of the question but i'm not quite sure ive got the right answer. And i don't understand what it means by scalar parametric form in part b, does the question just want me to write it as three separate linear equations?

2. Apr 29, 2012

### tiny-tim

hi ronho1234!
yes

"scalar" simply means that you use an "obvious" parameter, instead of a (perfectly valid but) stupid :yuck: one …

eg x = t, y = 3t + 5

as opposed to x = t2cosht, y = 3t2cosht + 5

3. Apr 30, 2012

### HallsofIvy

Staff Emeritus
Well, I don't believe I agree with tiny-tim that it is a difference between "reasonable" and "stupid" parameterizations! both x= t, y= 3t+5 and $x= t^2cosh(t)$, $y= 3t^2 cosh(t)+ 5$ are perfectly good scalar parameterizations.

The point is that the give three scalar (numerical) equations for x and y as opposed to the vector equation $\vec{r}(x,y)= t\vec{i}+ (3t+ 5)\vec{j}$.

You understand, I hope, that these are NOT the solution to your problem stated above which is three dimensional. For that you need to know that the line through point $(x_0, y_0, z_0)$, perpendicular to plane Ax+ By+ CZ= D has scalar parametric equations $x= At+ x_0$, $y= Bt+ y_0$, $z= Ct+ z_0$.

4. Apr 30, 2012

### LCKurtz

But some parameterizations really are better than others in a given setting. Tiny Tim's second parameterization doesn't give the whole line that his first one does in his example.

5. May 1, 2012

### tiny-tim

i'm not convinced …

my guess is that, by "scalar", the question means "linear"​

(as in "scalar multiplication" )

6. May 1, 2012

### HallsofIvy

Staff Emeritus
No, scalar simply means "number".

(More generally, in linear algebra, a "scalar" is a member of the underlying field of the vector space.)