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LAST NAME : FIRST NAME : QUIZ 1, Version A : MATH 251, Section 506 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] 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