Obtain an equation of the plane in the form ##px+qy+rz=d##

chwala
Gold Member
Messages
2,825
Reaction score
413
Homework Statement
see attached
Relevant Equations
vectors
1712819096284.png

The solution is here;
1712819128866.png



Now to my comments,
From literature, the cross product of two vectors results into a vector in the same dimension. A pointer to me as i did not know the first step. With that in mind and using cross product, i have

##(1-1)i - (-1-1)j+(1+1)k =0i+2j +2k## as shown in ms attachment.

Now the second part is the reason of this post. My take on that is,

##r=(λ +μ)i + (μ-4-λ)j + (λ-3-μ)k =x+y+z##

Now

##λ +μ = 0, ⇒ λ = -μ##

Therefore,
##μ-4+μ=y##
##-μ-3-μ=z##

on adding the above two equations,

##-4+-3=y+z##
##-7=y+z##

unless there is a better approach or simpler...have a great day.
 
Physics news on Phys.org
The general form of the equation for a plane is
$$
\vec n \cdot \vec r = C,
$$
where ##C## is a constant and ##\vec n## is normal to the plane. You have first found ##\vec n## so that is fine, but after that you are inserting the general form of any parametrised point on the plane. This is unnecessarily complicated. You know that the expression above should equal the same constant ##C## regardless of the point so using a single point in the plane will give you ##C##. The easiest is to just take the point corresponding to ##\mu =\lambda = 0##, i.e., ##\vec r = -4\hat j - 3\hat k##. Doing so with ##\vec n = \hat j + \hat k## results in
$$
C = -4 \cdot 1 - 3 \cdot 1 = -7
$$
and therefore
$$
\vec n \cdot \vec r = y + z = -7
$$
 
  • Informative
  • Like
Likes docnet and chwala
Also note that there is no requirement that ##\mu + \lambda = 0##, but this just describes a line in the plane. The values of ##\mu## and ##\lambda## are arbitrary, but the point is that you will get the same constant ##C## regardless of the values of ##\mu## and ##\lambda##.
 
This is also related part of question,

1712822314717.png


The solution is here - quite clear to me

1712822345388.png


but nothing wrong with me having,
##y+z-7=0##
##-5x+3y+5z-4=0##

setting ##x=0## gives

##y=-\dfrac{39}{2}## and ##z=\dfrac{25}{2}##

Thus,

##r+λ(n_1 ×n_2) = (0, -19.5, 12.5) +λ(2,-5,5)##
 
Sure, it is just a different parametrisation. Setting ##\lambda = \lambda' - 7/2## should give you the parametrisation given in the answer key.
 
This last part is also related to the question- hmmmm i have no idea how they did this but i will still share my thoughts.

1712826871823.png


1712826891897.png



Mythoughts,

Distance between a point and a plane is given by,

##D=\dfrac{|ax_0 +by_o+cz_0 +d|}{\sqrt{a^2+b^2+c^2}}=\sqrt{2}##

Now could they have used ,

##x=a +λ (1-b-a)##

##y=a+λ(b-a)##

##z=(a-7)+λ(b-a-7)##

when ##λ=0## I will then have,

##D=\dfrac{|0 +a+a-7+7|}{|\sqrt{0^2+1^2+1^2}|}=\sqrt{2}##

If that is the case then its time for me to have hot coffee! :cool:
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...

Similar threads

Back
Top