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QUIZ 1, Version B : MATH 251, Section 506
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Write up your result, detail your calculations if necessary and BOX your final answer
1. [25pts] Find the equation of the sphere with center (0, -1, 2) and radius
√
2.
2. [25pts] Find a unit vector which is orthogonal to both 2j + k and i + 2k.
3. [25pts] Determine whether the vectors v1 = h1, −1, 0i and v2 = h−1, −1, 4i are orthogonal, parallel or
neither.
4. [25pts] Find parametric equations and symmetric equations for the line passing through A (3, 1, −1)
and B (3, 2, 6).
1. The equation is x2 + (y + 1)2 + (z − 2)2 = 2 .
2. Denote v = 2j +k = h0, 2, 1i and w = i+2k = h1, 0, 2i. The cross product of both vectors is orthogonal
to both of them :
* 0 + * 1 + 2 0 0 1 0 1
= h4, 1, −2i.
2 × 0
=
+ (−1) +
1 2
1 2 2 0 1
2
√
√
The length of this vector is 4 + 16 + 1 = 21. Then, a unit vector orthogonal to both vectors is :
1
−2
4
√ ,√ ,√
.
21 21 21
3. We remark that v2 • v1 = 0 so these vectors are orthogonal.
~ = h3 − 3, 2 − 1, 6 − (−1)i = h0, 1, 7i. Let t ∈ R and (x, y, z) a
4. The line is directed by the vector AB
point of the line, parametric equations of the line are :
* x + * 3
+ * 0 +
* 3
+
y
1
1+t
=
+ 1 .t =
7
−1 + 7t
z
(−1)
Parametric equations are : hx, y, zi = h3, 1 + t, −1 + 7ti.
Symmetric equations are obtained from the previous equations :
t=y−1=
z+1
,
7
x = 3.