# 10 Calculus Problems

Please see the attached file for the fully formatted problems.

Problems involve:

parametric equation of line segment,

volume of a parallelipiped,

sketching a plane given the equation,

finding rectangular equations,

center and radius of a sphere using the equation of a sphere,

force vector problems.

https://brainmass.com/math/basic-calculus/parametric-equation-line-segments-8537

#### Solution Preview

1.)

The important points are: intersection of plane with 3 axes x, y, z.

x+2y+4z = 8

Interception on x axis: y=0,z=0, x = 8

Interception with y axis: x=0,z=0, y = 4

Interception with zaxis: x=0,y=0, z = 2

therefore,

three well known points are: (8,0,0), (0,4,0), (0,0,2)

See the attached file.

2.)

P(2,-1,4), Q(5,3,3)

x1-x2 = 2 - 5 = -3

y1-y2 = -1 - 3 = -4

z1-z2 = 4 - 3 = 1

therefore direction ratios of the line given as:

a:b:c = -3:-4:1

hence the parametric equation of straight line is given as:

x = x1 + a*r

y = y1 + b*r

z = z1 + c*r

=> (x = 2 - 3r, y = -1 - 4r, z = 4 + r) --Answer

3.)

postion vectors of P,Q,R:

P = (1,-1,2)=> vector OP = v(OP) = i - j + 2k

Q = (4,1,3) => v(OQ) = 4i + j + 3k

R = (-1,1,-1) => v(OR) = -i + j - k

v(PQ) = v(OQ) - v(OP) = 3i + 2j + k

v(RQ) = v(OQ) - v(OR) = 5i + 0.j + 4k

direction cosines of the normal to the plane:

normal vector to the plane = v(n)

v(n) = v(PQ) X v(RQ) = 8i - 7j - 10k

therefore, equation of the plane:

{v(r) - v(OP)}.v(n) = 0

because, r vector - OP vector will be in the plane and dot product ...

#### Solution Summary

The parametric equations of line segments are examined. A plane for an equation is sketched. Ten calculus problems are solved in detail.