For what value of c are the vectors [3, 1, c] and [-1,2, -1] orthogonal? 2....

 

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For what value of c are the vectors [3, 1, c] and [-1,2, -1] orthogonal? 2. 15 pts. Use the cross product to find the area of the parallelogram made by the points (-1, 2, 3), (4,2,3), (3, -1,3), and (-2, -1, 3). 3. 15 pts. Use the dot product to find the 4 interior angles of the paral- lelogram in the previous problem. 4. 10 pts. Find the unit surface normal vector to the surface 9x2 + y+ 422 14 at the point P: (1, 1, 1) = 5. 10 pts. For gey and u =[y, z, a] calculate the j-component of curl(gu). (The full curl calculation has i-, j- and k-components; this problem just asks for the j-component.) 6. 10 pts. Calculate div v and its value at P for v =[x cos x, x + y+ 22, x+ y+z2], P: (0,3,4). 7. 15 pts (5 pts each for (a), (b), and (c)). For the curve r(t) =[5 cost, 3 sint, -1] and the point P: (2-1) find (a) the tangent vector at P, (b) the unit tangent vector at P, and (c) the equation of the line tangent to the curve and passing through P. 8. 15pts. In one alternative to polar coordinates, we write x and y in terms of the two parameters r and 0, with xr sin and y = r cos (note that this is switched from the polar coordinates homework prob- lem). Let f(x, y) = 4xy. Use the multi-variable chain rule to find of and For what value of c are the vectors [3, 1, c] and [-1,2, -1] orthogonal? 2. 15 pts. Use the cross product to find the area of the parallelogram made by the points (-1, 2, 3), (4,2,3), (3, -1,3), and (-2, -1, 3). 3. 15 pts. Use the dot product to find the 4 interior angles of the paral- lelogram in the previous problem. 4. 10 pts. Find the unit surface normal vector to the surface 9x2 + y+ 422 14 at the point P: (1, 1, 1) = 5. 10 pts. For gey and u =[y, z, a] calculate the j-component of curl(gu). (The full curl calculation has i-, j- and k-components; this problem just asks for the j-component.) 6. 10 pts. Calculate div v and its value at P for v =[x cos x, x + y+ 22, x+ y+z2], P: (0,3,4). 7. 15 pts (5 pts each for (a), (b), and (c)). For the curve r(t) =[5 cost, 3 sint, -1] and the point P: (2-1) find (a) the tangent vector at P, (b) the unit tangent vector at P, and (c) the equation of the line tangent to the curve and passing through P. 8. 15pts. In one alternative to polar coordinates, we write x and y in terms of the two parameters r and 0, with xr sin and y = r cos (note that this is switched from the polar coordinates homework prob- lem). Let f(x, y) = 4xy. Use the multi-variable chain rule to find of and

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