Line and Plane: Perpendicular or Parallel? Explanation and Examples

[multicalcprob]

Can someone explain to me how to do this:

Determine whether the line and plane are perpendicular or parallel or neither
x = -1+2t
y = 4+t
z = 1-t

4x+2y-2z-7=0

My attempt:
2/4 = 1/2 = -1/-2
Since the ratios are the same, does it mean it is parallel?

Also when is it perpendicular?

 

[weatherhead]

Do you have any experience in using vectors to tackle problems like this? Your method lacks the finesse needed t be sure of these things in 3 dimensions.
1) do you know how to find the direction vector of the normal to the plane?
2) can you find a direction vector of the line?
3)do you know how to find the angle between these vectors (hint: dot product)

ok there are other ways to do this problem, so ignore me if you hate vectors, I'm just a bit of a junkie for them

 

[multicalcprob]

So parallel is wrong?

 

[weatherhead]

haven't worked it out, but I can tell you that if it is, you got that answer by a lucky guess. That just isn't a good way to approach a 3d problem. Give me two minutes I'll tell you if you're right or not... goes to get paper...

 

[weatherhead]

... they aren't parallel

 

[weatherhead]

I'll give you a clue, the plane is perpendicular to the line

 

[HallsofIvy]

Remember that the vector <2, 1, -1> points in the same direction as the line x = -1+2t,
y = 4+t, z = 1-t but that the vector <4, 2, -2> is perpendicular to the plane
4x+2y-2z-7=0. Having shown that those two vectors are parallel, it follows that the line and plane are perpendicular.

 

[multicalcprob]

that makes sense i solved the other 3 problems the same way thanks for your help

 

[natives]

The bellow equations are parametric representations of vectors where as:
x = -1+2t
y = 4+t
z = 1-t
are parametric representations of a vectorial line
and
4x+2y-2z-7=0
is a parametric representation of a plane...

How does that occur and what's the concept to determine whether the line and plane are parallel,perpendicular or otherwise is as below...

A line has the general vector representation:

r= (p₁,p₂,p₃)+L (d₁,d₂,d₃)

in your case the letter L above is the t in the equations thus
(d₁,d₂,d₃)
=(2,1,-1) (its tiresome to get how I get this but if you knew a bit about vectors you'd definitely know how that came about!)...this is called the directional vector of the line and can be used to determine an answer in your question.
and (p₁,p₂,p₃) is just a point on the line...


A plane has the general representation as:

ax+by+cz=k

where the constants (a,b,c)=(4,2,-2) are constants in the normal vector of the plane...

All you have to do is do the dot product of the two vectors such that if the dot product is zero then the vectors are perpendicular and if it is equal to the multiple of the magnitudes of the two vectors then it is parallel and if it is equal to a fraction of the product of the magnitudes then there is an angle theta other than 0 or 90 between the 2 vectors...

the dot product is (2,1,-1).(4,2,-2)=8+2+2=12

and product of magnitudes is sqrroot(4+1+1)*sqrroot(16+4+4)=sqrroot(144)=12


since dot product and product of magnitudes is the same then the line is PARALLEL to the plane!

 

[natives]

The bellow equations are parametric representations of vectors where as:
x = -1+2t
y = 4+t
z = 1-t
are parametric representations of a vectorial line
and
4x+2y-2z-7=0
is a parametric representation of a plane...

How does that occur and what's the concept to determine whether the line and plane are parallel,perpendicular or otherwise is as below...

A line has the general vector representation:

r= (p₁,p₂,p₃)+L (d₁,d₂,d₃)

in your case the letter L above is the t in the equations thus
(d₁,d₂,d₃)
=(2,1,-1) (its tiresome to get how I get this but if you knew a bit about vectors you'd definitely know how that came about!)...this is called the directional vector of the line and can be used to determine an answer in your question.
and (p₁,p₂,p₃) is just a point on the line...


A plane has the general representation as:

ax+by+cz=k

where the constants (a,b,c)=(4,2,-2) are constants in the normal vector of the plane...

All you have to do is do the dot product of the two vectors such that if the dot product is zero then the vectors are perpendicular and if it is equal to the multiple of the magnitudes of the two vectors then it is parallel and if it is equal to a fraction of the product of the magnitudes then there is an angle theta other than 0 or 90 between the 2 vectors...

the dot product is (2,1,-1).(4,2,-2)=8+2+2=12

and product of magnitudes is sqrroot(4+1+1)*sqrroot(16+4+4)=sqrroot(144)=12


since dot product and product of magnitudes is the same then the line is PARALLEL to the plane!