Question

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin. (a) (b) (c) (d)

Answer 1

The Cartesian equation of plane in normal form is expressed as,

lx+my+nz=d (1)

The coordinates of the foot of the perpendicular drawn from the origin are,

( ld l 2 + m 2 + n 2 , md l 2 + m 2 + n 2 , nd l 2 + m 2 + n 2 )

(a)

The Cartesian equation of the plane is,

2x+3y+4z−12=0 2x+3y+4z=12 (2)

Compare equation (1) and (2).

l=2

m=3

n=4

d=12

The coordinates of the foot of the perpendicular drawn from the origin are,

( 2×12 2 2 + 3 2 + 4 2 , 3×12 2 2 + 3 2 + 4 2 , 4×12 2 2 + 3 2 + 4 2 )

Or

( 24 4+9+16 , 36 4+9+16 , 48 4+9+16 )

Or

Thus, the coordinates of the foot of the perpendicular drawn from the origin are ( 24 29 , 36 29 , 48 29 ) .

(b)

The Cartesian equation of the plane is,

3y+4z−6=0 3y+4z=6 (3)

Compare equation (1) and (3).

l=0

m=3

n=4

d=6

The coordinates of the foot of the perpendicular drawn from the origin are,

( 0×6 0 2 + 3 2 + 4 2 , 3×6 0 2 + 3 2 + 4 2 , 4×6 0 2 + 3 2 + 4 2 )

Or

( 0, 18 9+16 , 24 9+16 )

Or

( 0, 18 25 , 24 25 )

Thus, the coordinates of the foot of the perpendicular drawn from the origin are ( 0, 18 25 , 24 25 ) .

(c)

The Cartesian equation of the plane is

x+y+z=1(4)

Compare equation (1) and (4).

l=1

m=1

n=1

d=1

The coordinates of the foot of the perpendicular drawn from the origin are

( 1×1 1 2 + 1 2 + 1 2 , 1×1 1 2 + 1 2 + 1 2 , 1×1 1 2 + 1 2 + 1 2 )

Or

( 1 3 , 1 3 , 1 3 )

Thus, the coordinates of the foot of the perpendicular drawn from the origin are ( 1 3 , 1 3 , 1 3 ) .

(d)

The Cartesian equation of the plane is,

5y+8=0 5y=−8 (5)

Compare equation (1) and (5).

l=0

m=5

n=0

d=−8

The coordinates of the foot of the perpendicular drawn from the origin are

( 0×−8 0 2 + 5 2 + 0 2 , 5×−8 0 2 + 5 2 + 0 2 , 0×−8 0 2 + 5 2 + 0 2 )

That is,

( 0, −8 5 ,0 )

Thus, the coordinates of the foot of the perpendicular drawn from the origin are ( 0, −8 5 ,0 ) .