Question

The fact that l₁ and l₂ do not intersect and are not parallel implies they are skew, which means that they lie in different parallel planes. Find equations for these two planes.

a) Ax + By + Cz = D₁, where A, B, and C are constants, and D₁ is a constant related to the first plane.
b) Ex + Fy + Gz = D₂, where E, F, and G are constants, and D₂ is a constant related to the second plane.
c) mx + ny + pz = Q₁, where m, n, and p are constants, and Q₁ is a constant related to the first plane.
d) qx + ry + sz = Q₂, where q, r, and s are constants, and Q₂ is a constant related to the second plane.

Answer 1

The equations for the two planes can be written in the form Ax + By + Cz = D, where A, B, C, and D are constants related to the planes. For the first plane, the constants are based on the direction vector of l₁. For the second plane, the constants are based on the direction vector of l₂. Option B is correct.

Step-by-step explanation:

The fact that l₁ and l₂ do not intersect and are not parallel implies they are skew, which means that they lie in different parallel planes. To find the equations for these two planes, we need to use the direction vectors of the lines.

Let's say that the direction vector of l₁ is v₁ = ⟨a₁, b₁, c₁⟩ and the direction vector of l₂ is v₂ = ⟨a₂, b₂, c₂⟩. The equation of a plane is in the form Ax + By + Cz = D.

For the first plane:

A = a₁

B = b₁

C = c₁

D₁ can be any constant. It can be chosen as one of the coordinates of a point on the plane.

So, the equation for the first plane is: Ax + By + Cz = D₁.

Similarly, for the second plane:

A = a₂

B = b₂

C = c₂

D₂ can be any constant. It can be chosen as one of the coordinates of a point on the plane.

So, the equation for the second plane is: Ex + Fy + Gz = D₂.