Stuck on vector calculus questions for exam

[terryfields]

two questions that i can't ever remember covering

first of all finding a perpendicular vector

find a vector perpendicular to the vectors a=i+2j-2k and b=-2i+3j+5k

and secondly the equation of a plane?? through point with position vector (2,1,1) and perpendicular to (3,-1,2) what are the forumla for finding these pieces of information?

 

[rootX]

Show some attempt

1) Use cross product
2) (3,-1,2) is perpendicular to the plane and (2,1,1) is a plane point

These both questions are really simple (you are not extending your information etc.). If you know your basics you should be able to solve them.

 

[terryfields]

so the first parts just the cross product? then to find the perp unit vector i divide it by its length?

 

[terryfields]

whats the equation of a plane? and are there different ways to work it out given different information because the notes i have on it don't seem to make much sense to me i was expecting some kind of formula but there doesn't seem to be one

 

[terryfields]

when given 3 points do i do the cross product of two distances to find the normal vector? e.g A,B,C (B-A)X(C-A)=N and if this is so what do i after I've found the normal?

 

[Defennder]

so the first parts just the cross product? then to find the perp unit vector i divide it by its length?

Yes correct.

whats the equation of a plane? and are there different ways to work it out given different information because the notes i have on it don't seem to make much sense to me i was expecting some kind of formula but there doesn't seem to be one

The equation of a plane is given by the dot product of a vector lying on the plane from a reference point on the plane and the vector normal to the plane.

when given 3 points do i do the cross product of two distances to find the normal vector? e.g A,B,C (B-A)X(C-A)=N and if this is so what do i after I've found the normal?

Yes this is correct. The rest is as explained above.

 

[terryfields]

anyone arround? could really use some help on this I've got to the point where i now know (or think i know) that Ax+By+Cz+D=0 is the equation of the plane with A,B, C being the normal vector and D being the distance from the plane to the orogin but i have no idea how to find these two peices of information from the information that i have been given in any of the examples, thanks

 

[Defennder]

That is the general equation of a plane. It is obtained by the means as I have explained to you above.

 

[terryfields]

and secondly the equation of a plane?? through point with position vector (2,1,1) and perpendicular to (3,-1,2)

so for this question am i correct in thinking that i need to find the line perpendicular to the perpendicular (i.e the normal) and then dot product with the other vector?

 

[terryfields]

but how do i find the perp of 3,-1,2 with only one vector to go on? i can't use the cross product this way as above?

 

[terryfields]

and for the third part i just use my normal along with anyone of the 3 points and cross product them? thanks for ur help so far defender

 

[rootX]

if n perp to plane

then
n dot any plane position = D (in standard eqn)

and

<a,b,c> = n

ax+by+cz = d

(I learned this in high school discrete math, maybe they don't teach well in calculus or assume that you know this already ..)

 

[BoundByAxioms]

Given a normal vector n, and a point r_o, the equation of a plane perpendicular to that vector and through that point is n\circ(r- r_o)=0. (That circle things represents a dot). If you are using Stewart's Calculus text, there is a pretty good section about this in that book.

 

[Josyulasharma]

For the first one use cross product so you'll get a vector perpendicular to both
second the d.c's of the plane are the cordinaates of the Vector perpendicular to the plane and it passes through the given point,subs and get the ans
ax+by+cz+d=0
a,b,c (you know them)
it passes through
a',b',c'
so
d=-(aa'+bb'+cc')

 

[Josyulasharma]

For the first one use cross product so you'll get a vector perpendicular to both
second the d.c's of the plane are the cordinaates of the Vector perpendicular to the plane and it passes through the given point,subs and get the ans
ax+by+cz+d=0
a,b,c (you know them)
it passes through
a',b',c'
so
d=-(aa'+bb'+cc')