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Answer:
SOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253
1. Find the equation of a sphere if one of its diameters has end points (1, 0, 5) and
(5, −4, 7).
Solution:
The length of the diameter is p
(5 − 1)2 + (−4 − 0)2 + (7 − 5)2 =
√
36 = 6, so the
radius is 3. The centre is at the midpoint ( 1+5
2
,
0−4
2
,
5+7
2
) = (3, −2, 6). Hence, the
sphere is given as (x − 3)2 + (y + 2)2 + (z − 6)2 = 9 .
2. Find vector, parametric, and symmetric equations of the following lines.
(a) the line passing through the points (3, 1,
1
2
) and (4, −3, 3)
Solution:
The vector between two points is ~v = h4 − 3, −3 − 1, 3 −
1
2
i = h1, −4,
5
2
i. Hence
the equation of the line is
Vector form: ~r = ~r0 + t~v = h4, −3, 3i + th1, −4,
5
2
i = h4 + t, −3 − 4t, 3 + 5
2
ti
Parametric form: x = 4 + t, y = −3 − 4t, z = 3 + 5
2
t
Step-by-step explanation: