Equation of Plane & Line Passing Through Points: Find E & Dist.

[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 I've 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?

 

[tiny-tim]

hi ronho1234!

(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.

… 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?

yes

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

eg x = t, y = 3t + 5

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

 

[HallsofIvy]

(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.

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.

 

[LCKurtz]

hi ronho1234!


yes

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

eg x = t, y = 3t + 5

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

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.

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.

 

[tiny-tim]

… both x= t, y= 3t+5 and x= t^2cosh(t), y= 3t^2 cosh(t)+ 5 are perfectly good scalar parameterizations.

i'm not convinced …

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

(as in "scalar multiplication" )

 

[HallsofIvy]

No, scalar simply means "number".

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