MATH 1260 - Quiz 1
Solution
(1) Find the unique value of c for which the lines R
1 intersect.
= ( t, − 6 t + c, 2 t − 8) and R
2
= (3 t + 1 , 2 t, 0)
Let P = ( x, y, z ) be the point where the lines intersect. There exists a time t such that the first line is at the point P at time t , and therefore

 x =
 t y = − 6 t + c z = 2 t − 8
.
Also, there exists a time s such that the second line is at the point P at time s , and therefore

 x = 3 s + 1 y = 2 s
 z = 0
.
Setting the two sets of equations equal to each other, we obtain that

 t = 3 s + 1

− 6 t + c = 2 s
2 t − 8 = 0
, which we can solve into
 t = 4
 s = 1
 c = 26
Therefore, c must be 26 for the two lines to intersect, and the point of intersection is (4 , 2 , 0).
(2) Find the distance from the point (1 , 1 , 1) to the plane x − y − z + 10 = 0.
The vector normal to the plane is h 1 , − 1 , − 1 i . The line through the point (1 , 1 , 1) with direction h 1 , − 1 , − 1 i has equations

 x = 1 + t
 y = 1 − t z = 1 − t
.
If we plug these into the equation of the plane, we obtain that
(1 + t ) − (1 − t ) − (1 − t ) + 10 = 0 , which gives t = − 3 .
Therefore, the line intersects the plane at time t = − 3 at the point ( − 2 , 4 , 4), and the distance from the point (1 , 1 , 1) to the plane is the same as the distance from the point (1 , 1 , 1) to the point
( − 2 , 4 , 4), which is
||h− 3 , 3 , 3 i|| =
√
9 + 9 + 9 =
√
27 .
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