MATH 1C De Anza College Symmetric Equation Exercises

I need it in two hour

All questions must have detailed answers.

(1) A line goes through the point (1, 3, ?2) and is orthogonal to the plane given by 2x ? y + z = 6.(a) Give a parametric equation of the line.(b) Give a symmetric equation of the line.(c) Determine at what point this line intersects the plane given by x ? y + z = ?12 .(2) Write an integral representing the length of the curve given byr = ?from ? = 0 to ? = ?. You do not need to evaluate the integral.(3) Find the area inside one loop of the curver = 5 sin 4?(4) Consider the parameterized curve given byx = t + ln ty = 2t3 + 3t2t > 0(a) Find dydx . Simplify your answer.(b) Find the equation of the tangent line to the curve at t = 1.(c) Find d2ydx2 . Simplify your answer.(5) Find parametric equations for a particle travelling clockwise around the circle centered at (5, ?2)with radius 3, starting at the point (2, ?2). Use a paramter t, 0 ? t ? 2?. Write your final answerin terms of cost and sin t.(6) Find an equation of the sphere that passes through the point (3, ?2, 4) and has center (1, ?1, 2).(7) Find an equation of the plane that passes through the points (4, 1, ?2), (?3, 3, ?1), and (2, ?4, 2).(8) Consider the vectors ~a = h4, ?2, ?4i and ~b = h2, 3, ?6i.(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(9) Find the area of the triangle in 3-dimensional space whose vertices are (1, 2, 3), (3, ?1, 2), and (1, 1, 1)(10) Find the distance from the point (1, 2, ?1) to the plane given by x ? 2y + 2z = 7(11) Find the volume of the parallelepiped determined by the vectors ~a,~b, and ~c where~a = h1, 2, 3i~b = h0, 3, 1i~c = h2, 1, 3i