-->

Vector Algebra

Question
CBSEENMA12035748

Find the vector equation of the plane which contains the line of intersection of the planes. straight r with rightwards arrow on top. open parentheses straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top close parentheses minus 4 space equals 0 and straight r with rightwards arrow on top. open parentheses 2 straight i with hat on top plus straight j with hat on top minus straight k with hat on top close parentheses plus 5 space equals 0 and which is perpendicular to the plane straight r with rightwards arrow on top. space open parentheses 5 straight i with hat on top plus 3 straight j with hat on top minus 6 straight k with hat on top close parentheses plus 8 space equals space 0

Solution

The equation of the given planes are
straight r with rightwards arrow on top. open parentheses straight i with hat on top plus 2 straight j plus 3 straight k with hat on top close parentheses minus 4 space equals space 0 space space... left parenthesis 1 right parenthesis
straight r with rightwards arrow on top. left parenthesis 2 straight i with hat on top plus straight j with hat on top minus straight k with hat on top right parenthesis plus 5 space equals space 0 space space space... left parenthesis 2 right parenthesis
The equation of the plane passing through the intersection of the planes (1) and (2) is
open square brackets straight r with rightwards arrow on top. open parentheses straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top close parentheses minus 4 close square brackets plus straight lambda open square brackets straight r with rightwards arrow on top. open parentheses 2 straight i with hat on top plus straight j with hat on top minus straight k with hat on top close parentheses plus 5 close square brackets equals 0
space space rightwards double arrow space space straight r with rightwards arrow on top open square brackets open parentheses 1 plus 2 straight lambda close parentheses straight i with hat on top plus left parenthesis 2 plus straight lambda right parenthesis straight j with hat on top plus left parenthesis 3 minus straight lambda right parenthesis straight k with hat on top close square brackets space equals space 4 minus 5 straight lambda space space space... left parenthesis 3 right parenthesis
Given that plane (3) is perpendicular to the plane straight r with rightwards arrow on top. open parentheses 5 straight i with hat on top plus 3 straight j with hat on top minus 6 straight k with hat on top close parentheses plus 8 space equals space 0
open parentheses 1 plus 2 straight lambda close parentheses cross times 5 plus left parenthesis 2 plus straight lambda right parenthesis cross times 3 plus left parenthesis 3 minus straight lambda right parenthesis cross times left parenthesis negative 6 right parenthesis space equals space 0
rightwards double arrow space space 19 straight lambda minus 7 space equals space 0
rightwards double arrow space space space straight lambda space equals space 7 over 19
Putting space straight lambda space equals space 7 over 19 space in space left parenthesis 3 right parenthesis comma space we space get
straight r with rightwards arrow on top open square brackets open parentheses 1 plus 14 over 19 close parentheses straight i with hat on top plus open parentheses 2 plus 7 over 19 close parentheses straight j with hat on top plus open parentheses 3 minus 7 over 19 close parentheses straight k with hat on top close square brackets space equals space 4 minus 35 over 19
rightwards double arrow space space straight r with rightwards arrow on top. open parentheses 33 over 19 straight i with hat on top plus 45 over 19 straight j with hat on top plus 50 over 19 straight k with hat on top close parentheses space equals space 41 over 19
rightwards double arrow space space straight r with rightwards arrow on top. open parentheses 33 straight i with hat on top plus 45 straight j with hat on top plus 50 straight k with hat on top close parentheses space equals space 41
This is the equation of the required line.