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QUIZ 1, Version A : 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 (1, -2, -1) and radius
√
3.
2. [25pts] Find a unit vector which is orthogonal to both 2i + j and i + 2k.
3. [25pts] Determine whether the vectors v1 = h1, −1, 2i and v2 = h2, −2, 4i are orthogonal, parallel or
neither.
4. [25pts] Find parametric equations and symmetric equations for the line passing through A (2, 1, 8)
and B (6, 0, 3).
1. The equation is (x − 1)2 + (y + 2)2 + (z + 1)2 = 3 .
2. Denote v = 2i+j = h2, 1, 0i and w = i+2k = h1, 0, 2i. The cross product of both vectors is orthogonal
to both of them :
* 2 + * 1 + 1 0
2 1 2 1
= h2, −4, −1i.
1 × 0
=
+ (−1) +
0 2
0 2 1 0
0
2
√
√
The length of this vector is 4 + 16 + 1 = 21. Then, a unit vector orthogonal to both vectors is :
−4 −1
2
√ ,√ ,√
.
21 21 21
3. We remark that v2 = 2v1 so these vectors are collinear i.e parallel in the general sense.
~ = h6 − 2, 0 − 1, 3 − 8i = h4, −1, −5i. 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 + * 2 + * 4 +
* 2 + 4t +
y
1 + −1 .t =
1−t
=
−5
8 − 5t
z
8
Parametric equations are :
hx, y, zi = h2 + 4t, 1 − t, 8 − 5ti .
Symmetric equations are obtained from the previous equations :
t=
x−2
8−z
=1−y =
.
4
5