find the vector equation of the plane passing through three points with position vectors i + j - 2k, 2i - Maths - Three Dimensional Geometry

Question

find the vector equation of the plane passing through three
points with position vectors i + j - 2k, 2i - j + k and i + 2j + k
Also find the coordinates of the point of intersection of the plain
and the line r = 3i - j - k + u(2i - 2j + k)

Deepali Gupta · Accepted Answer

position vectors of 3 points: i+j-2k, 2i-j+k,i+2j+kequation of the
plane passing through these points isx-1y-1z+22-1-1-11+21-12-11+2
=0x-1y-1z+21-23013 =0x-1-6-3-y-13-0+z+21 =0x-1-9-y-13+z+21
=0-9x+9-3y+3+z+2 = 0-9x-3y+z +14=09x+3y-z=14eq of the plane in
vector form = r→ (9i^+3j^-k^) = 14eqn of line in cartesian
plane =x-32=y+1-2=z+11=λany point on line is
2λ+3,-2λ-1,λ-1for point of intersection this point
must satisfy the eqn of plane also9(2λ+3)
+3(-2λ-1)-(λ-1)=1418λ+27-6λ-3-λ+1=1411λ+25=1411λ=-11λ=-1so
point of intersection is 1,1,-2

find the vector equation of the plane passing through three points with position vectors i + j - 2k, 2i - j + k and i + 2j + k. Also find the coordinates of the point of intersection of the plain and the line r = 3i - j - k + u(2i - 2j + k)