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QUIZ 1, Version B : MATH 251, Section 505
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”An Aggie does not lie, cheat or steal, or tolerate those who do”
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Write up your result, detail your calculations if necessary and BOX your final answer
1. [25pts] Show that the equation x2 +4x+y 2 +4y+z 2 = 0 is a sphere and determine its center and radius.
2. [25pts] Determine whether the vectors v1 = h1, −1, −1i and v2 = h2, −2, 4i are orthogonal, parallel or
neither (detail your computations).
3. [50pts] Find parametric equations and symmetric equations for the line passing through A (1, 0, −1)
and B (2, 1, 3).
1. x2 + 4x + y 2 + 4y + z 2 = 0 is equivalent to
x2 + 4x + y 2 + 4y + z 2 = 0
⇔
⇔
(x + 2)2 − 4 + (y + 2)2 − 4 + z 2 = 0
(x + 2)2 + (y + 2)2 + z 2 = 8
We recognize the equation of the sphere with center (−2, −2, 0) and radius
√
√
8 = 2 2.
2. We remark that the dot product v1 • v2 = 0 so v1 and v2 are orthogonal .
~ = h2 − 1, 1 − 0, 3 − (−1)i = h1, 1, 4i. Let t ∈ R and (x, y, z) a
3. The line is directed by the vector AB
point of the line, parametric equations of the line are :
* x + * 1 + * 1 +
* 1+t
+
y
0
t
=
+ 1 .t =
z
−1
4
−1 + 4t
Parametric equations are :
hx, y, zi = h1 + t, t, −1 + 4ti .
Symmetric equations are obtained from the previous equations :
t=y =x−1=
1+z
.
4