Question

Find the distance between two skew lines x-1/2 = y-3/2 = z and x-2/4 = y+1/1 = z-4/8 ​

Answer 1

Answer:

Explanation:

for the line, we will need

A) a point that it actually passes through, say →a, AND

B) a vector describing the direction in which it travels, say →b

....such that the line itself is

→r=→a+λ→b    □



For →a, we can simply choose the completely arbitrary point, so here we choose the point at which z = 0

This means that the plane equations, namely

π1:2x−2y+z=1

π2:2x+y−3z=3

become

2x−2y=1

2x+y=3

and these we solve as simultaneous equations to get

x=76,y=23 and of course z=0

so →a=⎛⎜ ⎜⎝76230⎞⎟ ⎟⎠

for →b we need to calculate the vector cross product of the normal vectors for π1 and π2.

In the drawing below, we are looking right down the line of intersection, and we get an idea as to why the cross product of the normals of the red and blue planes generates a third vector, perpendicular to the normal vectors, that defines the direction of the line of intersection.

Answer 2

Answer:

1) a point that it actually passes through, say -a

2) a vector describing the direction in which it travels, say -b

→r=→a+λ→b

For -a, we can simply choose the completely arbitrary point, so here we choose the point at which z = 0

This means that the plane equations, namely

π1:2x−2y+z=1

π2:2x+y−3z=3

2x−2y=1

2x+y=3

and these we solve as simultaneous equations to get

x=76,y=23 z=0

a=76230

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