Q. 295.0( 1 Vote )

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


Answer :

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 .


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