Show that the two lines x−1/2 = y−2/3 = z−3/4 and x−4/5 = y−1/2 intersect. Find also the point of intersection of these lines.

To determine if the two lines intersect, we first express each line in parametric form.

Parametric Equations of the Lines

The first line is given by:

• x - 1/2 = y - 2/3 = z - 3/4 = t

From this, we can write the parametric equations:

• x = 1/2 + t
• y = 2/3 + t
• z = 3/4 + t

The second line is given by:

• x - 4/5 = y - 1/2 = z = s

From this, we can write the parametric equations:

• x = 4/5 + s
• y = 1/2 + s
• z = s

Setting the Equations Equal

To find the intersection, we need to set the equations equal to each other:

• 1/2 + t = 4/5 + s
• 2/3 + t = 1/2 + s
• 3/4 + t = s

Solving the Equations

We can rearrange the first equation:

• t - s = 4/5 - 1/2

Calculating the right side:

• 4/5 - 1/2 = 8/10 - 5/10 = 3/10

Thus, we have:

• t - s = 3/10 (1)

Now, rearranging the second equation:

• t - s = 1/2 - 2/3

Calculating the right side:

• 1/2 - 2/3 = 3/6 - 4/6 = -1/6

Thus, we have:

• t - s = -1/6 (2)

Now we have two equations:

• (1): t - s = 3/10
• (2): t - s = -1/6

Since both equations cannot hold true simultaneously, we need to check the third equation:

• 3/4 + t = s

Finding Values of t and s

Substituting (1) into the third equation:

• s = 3/4 + (s + 3/10)

Rearranging gives:

• 3/4 = 3/10

This is not possible, indicating that the lines do not intersect.

Conclusion

The two lines do not intersect, as the equations derived from their parametric forms lead to contradictions. Therefore, there is no point of intersection.