Advertisements
Advertisements
Question
Show that the lines ` (x+1)/-3=(y-3)/2=(z+2)/1; ` are coplanar. Find the equation of the plane containing them.
Solution
Two lines
`(x-x_1)/a_1=(y-y_1)/b_1=(z-z_1)/c_1 and (x-x_2)/a_2=(y-y_2)/b_2=(z-z_2)/c_2`
are coplanar, if
`|[x_2-x_1,y_2-y_1,z_2-z_1],[a_1,b_1,c_1],[a_2,b_2,c_2]|`
Here,x1 = -1 y1 = 3 z1 = -2
x2 = 0 y2 = 7 z2 = -7
a1= -3 b1 = 2 c1 = 1
a2= 1 b2 = -3 c2 = 2
`therefore |[0-(-1),7-3,-7-(-2)],[-3,2,1],[1,-3,2]|`
`=|[1,4,-5],[-3,2,1],[1,-3,2]|`
=1(7)-4(-7)-5(7)
=0
The given lines are coplanar.
Equation of the plane, containing the given lines is
`|[x_2-x_1,y_2-y_1,z_2-z_1],[a_1,b_1,c_1],[a_2,b_2,c_2]|=0`
`|[x-x_1,y-y_1,z-z_1],[-3,2,1],[1,-3,2]|=0`
(x+1)(4+3)-(y-3)(-6-1)+(z+2)(9-2)=0
7x+7+7y-21+7z+14=0
7x+7y+7z=0
x+y+z=0
APPEARS IN
RELATED QUESTIONS
Show that four points A, B, C and D whose position vectors are
`4hati+5hatj+hatk,-hatj-hatk-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk)` respectively are coplanar.
The lines `(x - 2)/(1) = (y - 3)/(1) = (z - 4)/(-k) and (x - 1)/k = (y - 4)/(2) = (z - 5)/(1)` are coplnar if ______.
If the origin and the points P(2, -7, 5), Q(2, 3, - 5) and R(x, y, z) are co-planar, then ______
Show that the lines: `(1 - x)/2 = (y - 3)/4 = z/(-1)` and `(x - 4)/3 = (2y - 2)/(-4) = z - 1` are coplanar.
If lines `(x - 1)/2 = (y + 1)/3 = (z - 1)/4` and x – 3 = `(y - k)/2` = z intersect then the value of k is ______.
If the origin and the points P(2, 3, 4), Q(1, 2, 3) and R(x, y, z) are coplanar, then ______.
If the straight lines `(x - 1)/2 = (y + 1)/k = z/2` and `(x + 1)/5 = (y + 1)/2 = z/k` are coplanar, then the plane(s) containing these two lines is/are ______.
Lines `overliner = (hati + hatj - hatk) + λ(2hati - 2hatj + hatk)` and `overliner = (4hati - 3hatj + 2hatk) + μ(hati - 2hatj + 2hatk)` are coplanar. Find the equation of the plane determined by them.
