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Hello everyone,
Here's an exercise I have to solve:
Given an orthonormal basis (O;i,j,k) and two lines m and n whose respective parametric equations are:
x = 2+t x = 43t'
y = 32t (t belongs to R) and y = 58t' (t' belongs to R)
z = 5t z = 7t'
1)Show that Lines m and n are noncoplanar.(easy)
2) Show that point A(2,3,5) belongs to line m.
Let B be a point on line n.
Find the locus of point I, midpoint of segment AB, while B moves along line n.
3) Let M be a point of line m, and B be a point of line n.
Find the locus of the midpoint of segment MB while M and B move along lines m and n respectively.
The answers I gave:
2)By substitution in the equation of line m, we see that A belongs to m.
point A = (2,3,5) corresponds to t=0
point B has the form (4,5,7) + (3,8,1)t'
I = (3,4,6) + (3/2,4,1/2)t'
The result shows that the locus of points halfway between a given point and a line is a line.
3)point M = (2,3,5) + (+1,2,1)t
point B = (4,5,7) + (3,8,1)t'
Midpoint of MB: (3,4,6) + (1/2,1,1/2)t + (3/2,4,1/2)t'
The result shows that the locus of the midpoint of MB is a plane (but how to explain clearly to the teacher that it's a plane??)
Well, I hope that I answered correctly on the two parts.And my second question is:
why this difference between parts 2 and 3? why the locus in part 3 is not a line as well (as in part 2)? I mean that in the two parts we're finding the locus of a midpoint...

Thanks for helping.
Here's an exercise I have to solve:
Given an orthonormal basis (O;i,j,k) and two lines m and n whose respective parametric equations are:
x = 2+t x = 43t'
y = 32t (t belongs to R) and y = 58t' (t' belongs to R)
z = 5t z = 7t'
1)Show that Lines m and n are noncoplanar.(easy)
2) Show that point A(2,3,5) belongs to line m.
Let B be a point on line n.
Find the locus of point I, midpoint of segment AB, while B moves along line n.
3) Let M be a point of line m, and B be a point of line n.
Find the locus of the midpoint of segment MB while M and B move along lines m and n respectively.
The answers I gave:
2)By substitution in the equation of line m, we see that A belongs to m.
point A = (2,3,5) corresponds to t=0
point B has the form (4,5,7) + (3,8,1)t'
I = (3,4,6) + (3/2,4,1/2)t'
The result shows that the locus of points halfway between a given point and a line is a line.
3)point M = (2,3,5) + (+1,2,1)t
point B = (4,5,7) + (3,8,1)t'
Midpoint of MB: (3,4,6) + (1/2,1,1/2)t + (3/2,4,1/2)t'
The result shows that the locus of the midpoint of MB is a plane (but how to explain clearly to the teacher that it's a plane??)
Well, I hope that I answered correctly on the two parts.And my second question is:
why this difference between parts 2 and 3? why the locus in part 3 is not a line as well (as in part 2)? I mean that in the two parts we're finding the locus of a midpoint...

Thanks for helping.