De Anza College Perimeters and Vectors Calculus Questions

All questions must have detailed answers.

(1) Find equations for the following in 3 dimensions. Simplify as appropriate.(a) The plane parallel to the y-z plane going through the point (1, 2, 3).(b) The sphere with a diameter that has at its endpoints (1, 5, ?2) and (?3, 1, 2).(c) The set of all points whose distance from the line given by the pair of equations x = 2 andy = ?1 is 3.(d) The set of all points whose distance to the z axis is equal to their distance to the x-y plane.(2) Consider the vectors ~a = h3, 0, ?4i and ~b h2, ?1, 2i.(a) Find the vector 3~a ? 2~b(b) Find a unit vector pointing in the same direction as ~b.(c) Find ~a ·~b(d) Find the projection of ~a onto ~b (proj~b~a).(e) Find cos ?, where ? is the angle between ~a and ~b.(f) Find ~a ×~b(3) A child pulls a sled through the snow on a level path with a force of 60 N exterted at an angle of40? above the horizontal.(a) Find the horizontal and vertical components of the force. Give both exact answers, and decimalapproximations to one decimal place.(b) Find the work the child does on the sled in pulling it 10 meters. Write out both an exact answer,and a decimal approximation to one decimal place.(4) Find the work done by a forceF~ = 4~i ? 2~j + 3~k Non a particle as it moves from (1, 2, 3) to (0, 1, 0) (coordinates measured in meters).(5) Find the area of the parallelogram with vertices (1, 2, 5), (3, 1, 7), (0, 0, 0), and (2, ?1, 2).(6) Find the volume of the parallelepiped determined by the vectors h1, 2, 3i, h0, 1, 1i, and h1, ?1, 2i(7) A line passes through the points (1, 2, 5) and (2, 0, 3).(a) Write an equation of the line in vector form.(b) Write an equation of the line in parametric form.(c) Write an equation of the line in symmetric form.(8) A line passes through the points (3, 2, 3), (0, 1, 1), and (?1, 2, 1)(a) Write an equation of the plane.(b) What angle does the above plane make with respect to the plane x + y + z = 0? Give an exactanswer, as well as an approximate answer to the nearest degree.(c) What is the distance from the above plane to the point (3, 2, 1)?