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QUIZ 1 : MATH 251, Section 516
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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, -1) and radius
√
2.
2. [25pts] Find a unit vector which is orthogonal to both ~i + ~j and ~i + ~k.
3. [25pts] Find the cross product of h1, 2, −1i and h−2, −4, 2i.
4. [25pts] Find parametric equations and symmetric equations for the line passing through A (2, 1, 8)
and B (6, 0, 3).
Answers :
1. The equation of the sphere with center (0, 1, −1) and radius
√
2 is :
(x − 0)2 + (y − 1)2 + (z − (−1))2 =
√
2
2 ,
so,
x2 + (y − 1)2 + (z + 1)2 = 2.
2. There are 2 ways to obtain the result.
The shortest one : We saw that the cross product of 2 vectors is an orthogonal vector to both of them.
We have ~i + ~j = h1, 1, 0i and ~i + ~k = h1, 0, 1i.
* 1 + * 1 + 1 0
1 1 1 1
+ (−1) 1 × 0
= 0 1 + 1 0 = h1, −1, −1i.
0 1
0
1
√
1 −1 −1
√ ,√ ,√
.
3 3 3
Another way : Let hx, y, zi be a vector which is orthogonal to both previous vectors, it should verify
the following system :
hx, y, zi • h1, 1, 0i = 0
x+y =0
y = −x
⇔
⇔
hx, y, z) • h1, 0, 1i = 0
x+z =0
z = −x
The length of this vector is
3. Then, a unit vector orthogonal to both vectors is :
so, hx, y, zi = h1, −1, −1i.x. The vector h1, −1, −1i is orthogonal to both vectors. To obtain a unit
vector, we divide by its length, the wanted vector is
h1, −1, −1i
1 −1 −1
p
.
= √ ,√ ,√
3 3 3
12 + (−1)2 + (−1)2
3. We observe that h−2, −4, 2i = −2.h1, 2, −1i, so these vectors are collinear and its cross product is zero.
~ = 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 :
* 2 + 4t +
* x + * 2 + * 4 +
1−t
1 + −1 .t =
y
=
8 − 5t
−5
8
z
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