Consider the points P312 Q131 and R245 and S127 A find a vec

Consider the points P(3,1,2), Q(1,3,1) and R(2,4,5) and S(1,2,7)
A) find a vector perpendicular to pq and pr
b)find an equation of the plane with p,q,r
c) find the minimum distance between the plane in point b and s
d) compute the volume of the tetrahedron defined by p,q,r,s

Solution

a. the vector PQ is OQ-OP vector so OQ means the vector from origin to the point Q,

so we have (1-3,3-1,1-2) =(-2,2,-1)

the vector PR is (2-3,4-1,5-2) = (-1,3,3)

b. In the xyz-coordinate system, equations of planes have the form

ax + by + cz = d,

where a, b, c, and d are real numbers. The method below will show you how to find the values for a, b, c, and d if you know the coordinates of three points. First convert the three points into two vectors by subtracting one point from the other two.

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

PQ = (-2,2,-1)

PR = (-1,3,3)

now Find the cross product of the vectors found

which is is (9,7,-4)

and hence we have the normal vector to the plane ax + by + cz = d,

The coefficients a, b, and c of the planar equation are 9, 7, and -4

so we have 9x + 7y -4z = d,

and to find the d just plug in some point say (1,3,1)

hence we get d = 26

so we have the equation.

9x + 7y -4z = 26.

c. i dont know which point b u r talking about.c question is not clear

d. The volume of the tetrahedron is 1/6 times the absolute value of the matrix determinant.

matrix formrd by the method. Call the four vertices of the tetrahedron (a, b, c), (d, e, f), (g, h, i), and (p, q, r).create a 4-by-4 matrix in which the coordinate triples form the colums of the matrix, with a row of 1\'s appended at the bottom:

so we 5.1667   P(3,1,2), Q(1,3,1) and R(2,4,5) and S(1,2,7)

this is the matrix we have and the determinant can be found to be 31

hence 31/6 is 5.1667