Question

find the vector and cartesian equation of the line passing through the point (1,2,-4) and perpendicular to each line x-8/8=y+19/-16=z-10/7 and x-15/3=y+29/8=z-5/-5

Answer 1

Answer:

Vector: ( 24, 61, 112)

Line equation:  ( x - 1 ) / 24  =  ( y - 2 ) / 61  =  ( z + 4 ) / 112

Step-by-step explanation:

A vector along the line

(x-8)/8 = (y+19)/(-16) = (z-10)/7

is the vector (8, -16, 7).

A vector along the line

(x-15)/3 = (y+29)/8 = (z-5)/(-5)

is (3, 8, -5).

A vector perpendicular to both of these is their cross product

( (-16)(-5)-(7)(8), (7)(3)-(8)(-5), (8)(8)-(-16)(3) )

= ( 80 - 56, 21 + 40, 64 + 48 )

= ( 24, 61, 112).

This is then the direction of the line in question.

As the line passes through ( 1, 2, -4), the equation for the line is then

( x - 1 ) / 24  =  ( y - 2 ) / 61  =  ( z + 4 ) / 112.