Question

Show that the lines L₁ :x=2−t,y=2t,z=5+t and L₂

:x=1+4t,y=3−8t,z=5−4t are parallel and find the distan between them.

Answer 1

To show that the lines L₁ and L₂ are parallel, compare their direction vectors. The distance between the two lines can be found using the formula for the shortest distance between skew lines.

Step-by-step explanation:

To show that the lines L₁ and L₂ are parallel, we can compare their direction vectors. The direction vector of L₁ is [2, 2, 1] and the direction vector of L₂ is [4, -8, -4]. Notice that these two vectors are scalar multiples of each other, meaning they have the same direction but possibly different magnitudes. Therefore, the lines L₁ and L₂ are parallel.

To find the distance between the two lines, we can use the formula for the shortest distance between two skew lines. Let P₁(x₁, y₁, z₁) be a point on L₁ and let P₂(x₂, y₂, z₂) be a point on L₂. Then, the distance between the lines L₁ and L₂ is given by:

d = √((x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²)

Substituting the known values, we have:

d = √((1+4t-2+t)² + (3-8t-2t)² + (5-4t-5-t)²)

Simplifying further, we get:

d = √(9t² + 12t + 14)

Therefore, the distance between the lines L₁ and L₂ is √(9t² + 12t + 14).