CBSE Class 12-science Answered

Vector equations

Asked by alanpeter9611 | 17 Feb, 2019, 02:54: AM

Expert Answer

Given equations of planes are

x+2y+3z-4 = 0 .........................(1)

2x +y-z+5 = 0 ..........................(2)

Equation of plane which contains the line of intersection of (1) and (2) is given by

(x+2y+3z-4 ) + λ (2x +y-z+5 ) = 0

(1+2λ)x + (2+λ)y +(3-λ)z + 5λ-4 = 0 ........................(3)

If we write the above eqn. in intercept form

begin mathsize 12px style fraction numerator x over denominator begin display style fraction numerator 4 minus 5 lambda over denominator 1 plus 2 lambda end fraction end style end fraction space plus space fraction numerator y over denominator begin display style fraction numerator 4 minus 5 lambda over denominator 2 plus lambda end fraction end style end fraction plus fraction numerator z over denominator begin display style fraction numerator 4 minus 5 lambda over denominator 3 minus lambda end fraction end style end fraction space equals space 1 end style

since it is given that x-intercept is twice of z-intercept,

begin mathsize 12px style fraction numerator 4 minus 5 lambda over denominator 1 plus 2 lambda end fraction space equals space 2 fraction numerator 4 minus 5 lambda over denominator 3 minus lambda end fraction end style

............................(4)

from eqn.(4), we get λ = 1/5. By substituting λ = 1/5 in eqn.(3) and after simplification, eqn.(3) is written as

7x + 11y +14z = 15 ...........................(5)

Direction ratios of normal to the plane, which is parallel to the plane given by eqn.(5) are 7, 11 and 14

Vector equation of a plane passing through (2, 3, -1) and parallel to plane (5) is

begin mathsize 12px style open square brackets r with rightwards arrow on top space minus open parentheses 2 i with rightwards arrow on top space plus space 3 j with rightwards arrow on top space minus space k with rightwards arrow on top close parentheses close square brackets times open parentheses 7 i with rightwards arrow on top space plus space 11 j with rightwards arrow on top space plus space 14 k with rightwards arrow on top close parentheses space equals space 0 space r with rightwards arrow on top times open parentheses 7 i with rightwards arrow on top space plus space 11 j with rightwards arrow on top space plus space 14 k with rightwards arrow on top close parentheses space equals space open parentheses 2 i with rightwards arrow on top space plus space 3 j with rightwards arrow on top space minus space k with rightwards arrow on top close parentheses times open parentheses 7 i with rightwards arrow on top space plus space 11 j with rightwards arrow on top space plus space 14 k with rightwards arrow on top close parentheses space r with rightwards arrow on top times open parentheses 7 i with rightwards arrow on top space plus space 11 j with rightwards arrow on top space plus space 14 k with rightwards arrow on top close parentheses space equals space 33 space................... left parenthesis 6 right parenthesis end style

eqn.(6) is the required vector equation

Answered by Thiyagarajan K | 18 Feb, 2019, 01:05: AM

Application Videos