Question #240207

a) Give parametric equation (point-direction form) of the line which lies on both of the planes:

x + y + z = 1 and d x + 2y + 10z = 2. What is the direction d of this line?

b) Let n1 and n2 be the normal vectors to the two given planes. Without actual computation,

describe the relationship between d and n1 × n2.


1
Expert's answer
2021-10-05T13:42:59-0400

a) if Ax + Bx +Cz =D is the equation of a plane , then the vector normal to the given plane is n1=i+2jkn_1 =i + 2 j -k

Equation of f2 is 3xy+4z=23x-y+4z=2

Therefore the normal vector n2=3ij+4kn_2= 3i - j +4k


b) We have n1=i+2jkn_1 =i + 2 j -k and n2=3ij+4k.n_2= 3i - j +4k.

Then n1n_1 and n2n_2 are perpendicular to the line of intersection. Therefore v=n1n2v = n_1 * n_2 is the perpendicular to both n1n_1 and n2n_2 and hence parallel to the plane line of intersection

[ijk121314]=i(81)j(4+3)+k(16)=7i7j7k\begin{bmatrix} i & j & k \\ 1 & 2 & -1\\ 3 & -1 & 4\\ \end{bmatrix}= i(8-1)-j(4+3)+k(-1-6)=7i-7j-7k


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