Math 225: Calculus III	Quiz 2    1/28&30/96

<<Calculus`Math225`;
i = {1,0,0}; j = {0,1,0}; k = {0,0,1};

1. Determine the parametric equations of the line through the points
{1,-2,4} and {-3,5,10}.

Solution

The direction of the line is given by the difference of the the two points:

p = {1,-2,4};
q = {-3,5,10};

v = q - p

To equations of the line through p = {1,-2,4} in the direction of v is:

{1,-2,4} + t v

x == 1 - 4t; y == -2 + 7t; z == 4 + 6t;

2.  Find the equation of the plane through the point {1,2,-1} that is
parallel to the vectors v = i - j + k and w = 2 i - k.

Solution

The cross product of v and w is perpendicular to the plane and therefore can be used for its normal vector:

n  =  Cross[i - j + k, 2  i - k]

The point {1,2,-1} is on the plane, so the equation of the plane is:

n.{x, y, z}  ==  n.{1, 2, -1}