Question -
Answer -
(a)┬а2x + 3y + 4z тАУ 12 = 0
Let the coordinate ofthe foot of┬атКе┬аP from theorigin to the given plane be P(x, y, z).
2x + 3y + 4z = 12 тАж.(1)
Direction ratio are(2, 3, 4)
тИЪ[(2)2┬а+(3)2┬а+ (4)2] =┬атИЪ(4 + 9 + 16)
=┬атИЪ29
Now,
Divide both the sidesof equation (1) by┬атИЪ29, weget
2x/(тИЪ29) + 3y/(тИЪ29) + 4z/(тИЪ29) =12/тИЪ29
So this is of the formlx + my + nz = d
Where, l, m, n are thedirection cosines and d is the distance
тИ┤ The direction cosinesare 2/тИЪ29, 3/тИЪ29, 4/тИЪ29
Coordinate of the foot(ld, md, nd) =
= [(2/тИЪ29) (12/тИЪ29), (3/тИЪ29) (12/тИЪ29), (4/тИЪ29) (12/тИЪ29)]
= 24/29, 36/29, 48/29
(b)┬а3y + 4z тАУ 6 = 0
Let the coordinate ofthe foot of┬атКе┬аP from theorigin to the given plane be P(x, y, z).
0x + 3y + 4z = 6 тАж.(1)
Direction ratio are(0, 3, 4)
тИЪ[(0)2┬а+(3)2┬а+ (4)2] =┬атИЪ(0 + 9 + 16)
=┬атИЪ25
= 5
Now,
Divide both the sidesof equation (1) by 5, we get
0x/(5) + 3y/(5) +4z/(5) = 6/5
So this is of the formlx + my + nz = d
Where, l, m, n are thedirection cosines and d is the distance
тИ┤ The direction cosinesare 0/5, 3/5, 4/5
Coordinate of the foot(ld, md, nd) =
= [(0/5) (6/5), (3/5)(6/5), (4/5) (6/5)]
= 0, 18/25, 24/25
(c)┬аx + y + z = 1
Let the coordinate ofthe foot of┬атКе┬аP from theorigin to the given plane be P(x, y, z).
x + y + z = 1 тАж. (1)
Direction ratio are(1, 1, 1)
тИЪ[(1)2┬а+(1)2┬а+ (1)2] =┬атИЪ(1 + 1 + 1)
=┬атИЪ3
Now,
Divide both the sidesof equation (1) by┬атИЪ3, weget
1x/(тИЪ3) + 1y/(тИЪ3) + 1z/(тИЪ3) = 1/тИЪ3
So this is of the formlx + my + nz = d
Where, l, m, n are thedirection cosines and d is the distance
тИ┤ The direction cosinesare 1/тИЪ3, 1/тИЪ3, 1/тИЪ3
Coordinate of the foot(ld, md, nd) =
= [(1/тИЪ3) (1/тИЪ3), (1/тИЪ3) (1/тИЪ3), (1/тИЪ3) (1/тИЪ3)]
= 1/3, 1/3, 1/3
(d)┬а5y + 8 = 0
Let the coordinate ofthe foot of┬атКе┬аP from theorigin to the given plane be P(x, y, z).
0x тАУ 5y + 0z = 8 тАж.(1)
Direction ratio are(0, -5, 0)
тИЪ[(0)2┬а+(-5)2┬а+ (0)2] =┬атИЪ(0 + 25 + 0)
=┬атИЪ25
= 5
Now,
Divide both the sidesof equation (1) by 5, we get
0x/(5) тАУ 5y/(5) +0z/(5) = 8/5
So this is of the formlx + my + nz = d
Where, l, m, n are thedirection cosines and d is the distance
тИ┤ The direction cosinesare 0, -1, 0
Coordinate of the foot(ld, md, nd) =
= [(0/5) (8/5), (-5/5)(8/5), (0/5) (8/5)]
= 0, -8/5, 0