Understanding Vector Calculus: Answer Check and Tips for Tomorrow's Exam

[terryfields]

i think i get this now but just checking as i have the exam tomorrow and won't do this part if I am getting it wrong.

(2) (a) find a vector perpendicular to a=i+2j-2k and b=-2i+3j+5k
just use the cross product? and get 15i+j+7k

(b) (i) find the equation of the plane through position vector (2,1,1) and perpendicular to the vector(3,-1,2)

is this Plane II=T(3,-1,2) + (2,1,1)

determin the perpendicular distance of plane II from the orogin

no idea here??

(ii) Find the equation of the plane III through three points (2,-2,1)=a (4,-1,6)=b and (3,-3,2)=c i use a-b and a-c and then do the cross product of these two answers to get (6i,-3j,-3k) am i right in thinking that it doesn't matter which 2 i use to get this cross product e.g i could use b-c b-a etc to find the normal then i have the answer planeIII=T(6,-3,-3)+(2,-2,1) (plus any point)

(iii) find the angles between the 2 planes, i'll wait for some feedback before i attempt this, thanks

 

[Defennder]

i think i get this now but just checking as i have the exam tomorrow and won't do this part if I am getting it wrong.

(2) (a) find a vector perpendicular to a=i+2j-2k and b=-2i+3j+5k
just use the cross product? and get 15i+j+7k

Yes correct.

(b) (i) find the equation of the plane through position vector (2,1,1) and perpendicular to the vector(3,-1,2)

is this Plane II=T(3,-1,2) + (2,1,1)

determin the perpendicular distance of plane II from the orogin

no idea here??

The equation of any plane is given by $$\vec{n} \cdot \vec{P_0 P} = 0$$ where n is the normal vector to the plane, OP is the position vector (x,y,z) and P0 is any reference point on the plane. The perpendicular distance from the plane can be found by assuming that the normal vector can be scaled by some constant so that it "touches" the origin.

(ii) Find the equation of the plane III through three points (2,-2,1)=a (4,-1,6)=b and (3,-3,2)=c i use a-b and a-c and then do the cross product of these two answers to get (6i,-3j,-3k) am i right in thinking that it doesn't matter which 2 i use to get this cross product e.g i could use b-c b-a etc to find the normal then i have the answer planeIII=T(6,-3,-3)+(2,-2,1) (plus any point)

You're right it does not matter, but make sure the two vectors you use are not parallel or anti-parallel (why?).

 

[terryfields]

sorry i don't understand this second part what is the vector P₀P? don't understand what i have to do with the normal vector here to find the equation of the plane?

 

[terryfields]

and so I've got the 2 plane equations wrong right?

 

[Defennder]

sorry i don't understand this second part what is the vector P₀P? don't understand what i have to do with the normal vector here to find the equation of the plane?

The P₀ simply refers to any point on the plane taken as a reference. For example if it is given that the point (1,2,3) lies on the plane, you can let P₀ be the reference point.

and so I've got the 2 plane equations wrong right?

That is equation of a line, if I interpret your T to mean the parameter which varies. The vector equation of a plane has two parameters, not one. You're not required to express the equation of a plane in vector form, are you? If not just stick to the x,y,z notation alone.

 

[terryfields]

so on this b(i) how do i go about finding (a) any point on the plane and (b) the normal vector as the only information i am given is a position vector (2,1,1,) and a perpendicular vector (3,-1,2)

 

[Defennder]

The position vector given is a point on the plane. As for the normal vector, what is definition of "normal"?

 

[terryfields]

but i have to do N.P0P doesn't that mean N dot product with the distance between the position vector and any point on the plane? if so isn't the distance between the position vector and itself going to be zero? as for the normal vector isn't that the opposite of the perpendicular? but i have no clue how to find it

 

[Defennder]

Yes it will equate to zero. That is how you get an equation. The normal vector to the plane is also perpendicular to the plane.