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LAST NAME : FIRST NAME : QUIZ 1, Version B : 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 (0, -1, 2) and radius √ 2. 2. [25pts] Find a unit vector which is orthogonal to both 2j + k and i + 2k. 3. [25pts] Determine whether the vectors v1 = h1, −1, 0i and v2 = h−1, −1, 4i are orthogonal, parallel or neither. 4. [25pts] Find parametric equations and symmetric equations for the line passing through A (3, 1, −1) and B (3, 2, 6). 1. The equation is x2 + (y + 1)2 + (z − 2)2 = 2 . 2. Denote v = 2j +k = h0, 2, 1i and w = i+2k = h1, 0, 2i. The cross product of both vectors is orthogonal to both of them : * 0 + * 1 + 2 0 0 1 0 1 = h4, 1, −2i. 2 × 0 = + (−1) + 1 2 1 2 2 0 1 2 √ √ The length of this vector is 4 + 16 + 1 = 21. Then, a unit vector orthogonal to both vectors is : 1 −2 4 √ ,√ ,√ . 21 21 21 3. We remark that v2 • v1 = 0 so these vectors are orthogonal. ~ = h3 − 3, 2 − 1, 6 − (−1)i = h0, 1, 7i. 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 + * 3 + * 0 + * 3 + y 1 1+t = + 1 .t = 7 −1 + 7t z (−1) Parametric equations are : hx, y, zi = h3, 1 + t, −1 + 7ti. Symmetric equations are obtained from the previous equations : t=y−1= z+1 , 7 x = 3.