12 grade maths others> Find the shortest distance between the li...

Find the shortest distance between the lines given by

r = (8 + 3λ)ˆi − (9 + 16λ)ˆj + (10 + 7λ)ˆk

r = 15ˆi + 29ˆj + 5ˆk + μ(3ˆi + 8ˆj − 5ˆk)

Aniket Singh , 8 Months ago

Grade

1 Answers

Askiitians Tutor Team

To find the shortest distance between the two lines given by the vector equations, we can use the formula for the distance between skew lines. The lines are represented as:

Line Equations

Line 1: r1 = (8 + 3λ) i - (9 + 16λ) j + (10 + 7λ) k
Line 2: r2 = (15 + 3μ) i + (29 + 8μ) j + (5 - 5μ) k

Direction Vectors

The direction vector for Line 1 is d1 = (3, -16, 7) and for Line 2 is d2 = (3, 8, -5).

Finding the Shortest Distance

The formula for the distance D between two skew lines is given by:

D = |(P2 - P1) · (d1 × d2)| / |d1 × d2|

Here, P1 is a point on Line 1 (e.g., (8, -9, 10)) and P2 is a point on Line 2 (e.g., (15, 29, 5)).

Calculating the Cross Product

First, we calculate the cross product d1 × d2:

d1 × d2 = |i j k|

|3 -16 7|

|3 8 -5|

This results in a vector that can be computed using the determinant.

Final Steps

After finding the cross product, substitute it back into the distance formula along with the vector (P2 - P1). Calculate the dot product and the magnitude of the cross product to find the shortest distance.

By following these steps, you will arrive at the shortest distance between the two lines. If you need further assistance with the calculations, feel free to ask!