Write the vector equations of the following lines and hence determine the distance between them:

We need to find the vector equations of the given lines and also determine the distance between them.

Let line be,

So, points (x, y, z) are on L₁.

Let us find (x, y, z).

⇒ (x – 1) = 2λ, (y – 2) = 3λ, (z + 4) = 6λ

⇒ x = 2λ + 1, y = 3λ + 2, z = 6λ – 4

So, the points on line L₁ comes out be (2λ + 1, 3λ + 2, 6λ – 4).

And the other line be,

So, points (x, y, z) are on L₂.

Let us find (x, y, z).

⇒ (x – 3) = 4μ, (y – 3) = 6μ, (z + 5) = 12μ

⇒ x = 4μ + 3, y = 6μ + 3, z = 12μ – 5

So, the points on line L₂ comes out to be (4μ + 3, 6μ + 3, 12μ – 5).

Let us find the vector equation of line L₁ using the points (2λ + 1, 3λ + 2, 6λ – 4):

Rearranging them,

…(i)

Now, let us find the vector equation of line L₂ usinf the points (4μ + 3, 6μ + 3, 12μ – 5):

Rearranging them,

…(ii)

We have got the vector equations namely, and .

Let us find the distance between the lines.

From (i),

Let,

From (ii),

Let,

Distance between the vector equations of the two given lines when is given by,

So, using the values of and , we get

Now, solving for . We get

Taking mod on both sides,

Let us find the mod of .

So,

Thus, vector equation of line L₁ is and line L₂ is and the distance between the lines is .