# Vector calculus and linear algebra

Question 4

Find the work done by the force

F (x,y,z) = -x^2y^3 i + 4j + xk

on moving charged electric particle along the path given by the equation

r (t) = 2cos t i + 2sintj + 4k,

where the parameter t varies from pi/4 to 7pi/4.

Question 5

Displacement of the spring system with friction is described by the differential equation

m d^2y / dt ^ 2 + c dy/dt + ky = 0,

where y is the displacement, m the mass, c the friction coefficient, and k the spring constant (all quantities are non-dimensional). Consider the case c^2 = 4mk (called critical)

a) Find the general solution y(t)

b) Find and sketch the solution satisfying the initial conditions y(0) = 2, y'(0) = -7

Question 7

Consider the vectors

a = (0,3,-2,1,4); b = (5,2,1,0,-1); c = (7, -3,6,21,0)

a) Find the length of the vector v = 2a - b;

b) Are any of the given three vectors parallel or orthogonal? Indicate which (if any)

See attachment for remaining questions and proper notation.

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#### Solution Preview

In physics, the work is given by the vector line integral of a force along a ...

#### Solution Summary

The solution solves for the length of vectors, general solutions of differential equations, and solutions satisfying the initial given conditions.