What is the intersecting line of two planes?

If ... ℓ,m,n ... are the directions Cosines of the

line of intersection L of the given two planes, then

5ℓ - 2m + 5n = 0,

-2ℓ +1m + 4n = 0.

Hence, using determinants,

ℓ / [ (-2)(4) - (5)(1) ] = - m / [ (5)(4) - (5)(-2) ] = n / [ (5)(1) - (-2)(-2) ]

ℓ / (-13) = m / (-30) = n / (1).

Then, the direction Ratios of line L are -13, -30, 1, i.e., 13, 30, -1. ... (1)

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Now, putting z = 0 in the eqs of the planes,

5x - 2y = -35 ... (2) ... and ... -2x + y = -11 ... (3)

2 x EQ(3) : - 4x + 2y = -22 ... (4)

EQ(2) + EQ(4) : x = -57

From (3), then : y = 2x - 11 = 2(-57)-11 = -125

That is, point P( -57, -125, 0 ) lies on line L ............ (5)

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From (1) and (5), eqs of line in symmetric form are

( x + 57 ) / 13 = ( y + 125 ) / 30 = ( z - 0 ) / (-1) = t.

Hence, its parametric eqs are

x = 13t - 57, y = 30t - 125, z = -t. .................. Ans.

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................... Happy To Help !

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The given system is

........(1)...5x - 2y + 5z = -35

........(2)...-2x + y + 4z = -11

First, obtain an equation (3) in terms of x and z only by multiplying the second equation by 2 and adding it to the first equation.

........(3)...x + 13z = -57

Let z = -t, and solve for x to get x = 13t - 57. Back-substitute the expressions obtained for x and z in equation (2) to get y = 2x - 4z -11 = 2(13t - 57) - 4(-t) - 11 = 30t - 125.

This answer agrees with Hemant's.

OPTIONAL: If you would like visual confirmation, please see the picture on photobucket. The blue dots correspond to the points (-5, -5, -4) and (1.5, 10, -4.5), both of which (can be seen to lie) on the line of intersection of the given planes.