Question

Consider the line L(t)=⟨−1−t,3−4t,3−2t⟩. Then: L is to the plane 5x+3y=15 L is to the plane 14x−2y−3z=−9 L is to the plane 2x+8y+4z=−4 L is to the plane 16x+5y−18z=−93

Answer 1

The line L is parallel to the planes 5x + 3y = 15 and 14x - 2y - 3z = -9, but it is not parallel to the planes 2x + 8y + 4z = -4 and 16x + 5y - 18z = -93.

Step-by-step explanation:

The direction vector of the line L(t) = ⟨-1 - t, 3 - 4t, 3 - 2t⟩ is given by ⟨-1, -4, -2⟩. For two vectors to be parallel, their direction vectors must be proportional. In the case of L and the planes 5x + 3y = 15 and 14x - 2y - 3z = -9, their direction vectors are proportional, indicating that L is parallel to these planes.

However, for the planes 2x + 8y + 4z = -4 and 16x + 5y - 18z = -93, their normal vectors are not proportional to the direction vector of L. This indicates that L is not parallel to these planes.

In summary, the final answer is that the line L is parallel to the planes 5x + 3y = 15 and 14x - 2y - 3z = -9, but it is not parallel to the planes 2x + 8y + 4z = -4 and 16x + 5y - 18z = -93. The concept of parallelism is crucial in understanding the orientation and relationship between lines and planes in three-dimensional space.