Is 2i + j + 3k the Normal Vector of Plane CDPQ?

[songoku]

Homework Statement

The planes ABCD and EFGH are parallel to the horizontal surface represented by the xy-plane. The triangles APD and BQC are congruent isosceles triangles and the lengths of the pillars AE, BF, CG, and DH are of the same height. The equation of the plane ABQP is given by -2x-y+3z = 2 and the point P has position vector 4i + 2j + 4k.

Explain why plane CDPQ is perpendicular to 2i + j + 3k. Hence show that the equation of the plane CDPQ is 2x + y + 3z = 22

Relevant Equations

Equation of plane: r . n = c

I know the normal of plane ABQP is -2i - j + 3k but I don't know how to prove that 2i + j + 3k is the normal vector of plane CDPQ

Thanks

 

[Cuckoo Beats]

Hi, hope you are in the fullest of your sprits!
To find the normal, you can use the vector product
If you don't know what that is, then

where ||A|| and ||B|| are magnitudes of vectors A and B respectively

 

[Mark44]

Hi, hope you are in the fullest of your sprits!
To find the normal, you can use the vector product
If you don't know what that is, then

where ||A|| and ||B|| are magnitudes of vectors A and B respectively

The formula shown is somewhat unclear, as it looks like the last factor is $\sin (\theta n)$. In fact, the formula should be as follows:
$$\vec A \times \vec B = \left(|\vec A| |\vec B| \sin(\theta)\right) \vec n$$
where $\vec n$ is the unit vector that is perpendicular to both $\vec A$ and $\vec B$.

 

[SammyS]

Hi, hope you are in the fullest of your sprits!
…

Hello @Cuckoo Beats ,

 

[olivermsun]

I know the normal of plane ABQP is -2i - j + 3k but I don't know how to prove that 2i + j + 3k is the normal vector of plane CDPQ

One way to think about this problem is to look at the plan view.

If the normal vector to plane ABQP is -2i - j + 3k, what does that normal vector look like in the plan view?

What does the normal vector to CDPQ look like in the plan view? How that vector relate to the previous normal vector you drew? (Think about any kind of symmetries that should be there.)

 

[songoku]

I think I get the hint.

Thank you very much Cuckoo beats, Mark44, SammyS, olivermsun