The vector equation of any plane passing through the line of intersection of the planes r m ₁= q ₁ and r m ₂= q ₂ is given by r ( m ₁+ m ₂ )= q ₁+ q ₂ for R . The vector equation of a plane passing through the point 2 i -3 j + k and the line of intersection of the planes r. ( i -2 j +3 k )=5 and r (3 i + j -2 k )=7

Step-by-step solution

Given, planes are r ( i -2 j +3 k )=5 ...(i) and r (3 i + j -2 k )=7 ...(ii) Hence equation of plane passing through the line of intersection of the plane, (i) and plane (ii) is given by aligned & r ₁ ( m ₁+ m ₂ )=q₁+ q₂, R & r [( i -2 j +3 k )+ (3 i + j -2 k )]=5+ (7) & r [ i (1+3 )+ j (-2+ )+ k (3-2 )]=5+7 aligned ...(iii) aligned & but given that r =(2 i -3 j + k ) & 2(1+3 )+(-3)(-2+ )+(1)(3-2 )=5+7 & 11+ =5+7 6 =6 =1 . aligned putting =1 in equation (iii), we get aligned & r [ i (1+3)+ j (-2+1)+ k (3-2)]=5+7 &