LAST NAME :
FIRST NAME :
QUIZ 1, Version A : MATH 251, Section 505
last name : . . . . . . . .
first name : . . . . . . . .
GRADE : . . . . . . . .
”An Aggie does not lie, cheat or steal, or tolerate those who do”
signature : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Write up your result, detail your calculations if necessary and BOX your final answer
1. [25pts] Show that the equation x2 + y 2 + z 2 = x is a sphere and determine its center and radius.
2. [25pts] Find a unit vector which is orthogonal to both i + j and i + j + k.
3. [50pts] Find parametric equations and symmetric equations for the line passing through A (1, 0, −1)
and B (2, 1, 3).
1. x2 + y 2 + z 2 = x is equivalent to
x2 − x + y 2 + z 2 = 0
⇔
⇔
1
1
(x − )2 − + y 2 + z 2 = 0
2
4
1
1 2
(x − ) + y 2 + z 2 = .
2
4
We recognize the equation of the sphere with center ( 12 , 0, 0) and radius 21 .
2. We know that the cross product of 2 vectors is an orthogonal vector to both of them. We have
~i + ~j = h1, 1, 0i and ~i + ~j + ~k = h1, 1, 1i.
* 1 + * 1 + 1 1 1 1 1 1
+ (−1) 1 × 1
= 0 1 + 1 1 = h1, −1, 0i.
0 1
0
1
The length of this vector is
√
1 −1
2. Then, a unit vector orthogonal to both vectors is : h √ , √ , 0i .
2 2
~ = 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