Question
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Give a geometric description of the set of points in space whose coordinates satisfy the given pair of equations. x2 + y2= 16, z = 2 Choose the correct description. The circle with center (0, 0, 2) and radius 4, parallel to the xy-plane The line through (4, 4, 2), parallel to the z-axis The line that passes through the points (4, 0, 2) and (0, 4, 2) The circle with center (4, 4, 0) and radius 2, parallel to the xy-plane For the points P1(-5, 3, 1) and P2(5, l, 0), find the direction of P1P2rightarrow and the midpoint of line segment P1P2. The direction of P1P2rightarrow is ( )i + ( )j + ( )k. (Type exact answers, using radicals as needed.) The midpoint of P1P2 is ( , , ). (Type integers or simplified fractions.) Find the following for the vectors u=-5i + 4j + root7 k and v = 5i - 4 j - root7 k. v · u, |v|, and |u| the cosine of the angle between v and u the scalar component of u in the direction of v the vector projvu v · u = (Simplify your answer.) |v| = (Type an exact answer, using radicals as needed.) |u| = (Type an exact answer, using radicals as needed.) The cosine of the angle between v and u is . (Type an exact answer, using radicals as needed.) The scalar component of u in the direction of v is . (Type an exact answer, using radicals as needed.) projvu = ( )i + ( )j + ( )k (Type exact answers, using radicals as needed.) Find a vector of magnitude 3 in the direction of v = 12i - 9 k. The vector is ( )i + ( )j + ( )k. (Simplify your answer. Use integers or fractions for any numbers in the expression.)
Explanation / Answer
1)
here x^2 + y^2 = 16 is a circle with center (0, 0) and radius = 4
so, option A is the correct one.
2)
P1P2 = (5+5) i + (1-3)j + (0 -1)k = 10 i - 2 j - k
mid point = (0, (3+1)/2, (1+0)/2) = (0, 2, 1/2) = (0, 2, 0.5)
3)
v.u = 5*(-5) + (-4)*4 + (-sqrt(7))*sqrt(7)
= -25 - 16 - 7 = -48
|v| = sqrt(25 + 16 + 7) = sqrt(48)
|u| = sqrt(25 + 16 + 7) = sqrt(48)
cos(theta) = -48/[sqrt(48)*sqrt(48)] = -1
