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LAST NAME : FIRST NAME : QUIZ 1, Version B : 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 +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