1104Senior Secofdary3. Show that the line joining the points A(1, 1, 2

1.	1104Senior Secofdary3. Show that the line joining the points A(1, 1, 2) and B(3, 4, -2) isicin to the noint A(2, 1, 1) isperpendicular to the line joining the points C(0, 3, 2) and D(3, 5, 6).
Answer» For two lines to be perpendicular , the product of their direction rations should be 0 (a1a2+b1b2+c1c2=0(a1a2+b1b2+c1c2=0 Let the given point beA(1,−1,2)B(3,4,−2)A(1,−1,2)B(3,4,−2)andC(0,3,2)C(0,3,2)andD(3,5,6)D(3,5,6) The direction ratios of the line joinigAandBis(3,1),(4−(−1)),(−2−2)(3,1),(4−(−1)),(−2−2) On simplifying we get, (2,5,−4)(2,5,−4) The direction ratios of the line joinigCandDis(3−0),(5−3),(6−2)(3−0),(5−3),(6−2) On simplifying we get, (3,2,4)(3,2,4) For the linesABandCDto be perpendicular a1a2+b1b2+c1c2=0 On substituting fora1,b1,c1a1,b1,c1anda2,b2,c2a2,b2,c2we get, =(2×3)+(5×2)+(−4×4)=(2×3)+(5×2)+(−4×4) =6+10−16=6+10−16 =0=0 HenceAB⊥CD

1104Senior Secofdary3. Show that the line joining the points A(1, 1, 2) and B(3, 4, -2) isicin to the noint A(2, 1, 1) isperpendicular to the line joining the points C(0, 3, 2) and D(3, 5, 6).

Answer»

For two lines to be perpendicular , the product of their direction rations should be 0

(a1a2+b1b2+c1c2=0(a1a2+b1b2+c1c2=0

Let the given point beA(1,−1,2)B(3,4,−2)A(1,−1,2)B(3,4,−2)andC(0,3,2)C(0,3,2)andD(3,5,6)D(3,5,6)

The direction ratios of the line joinigAandBis(3,1),(4−(−1)),(−2−2)(3,1),(4−(−1)),(−2−2)

On simplifying we get,

(2,5,−4)(2,5,−4)

The direction ratios of the line joinigCandDis(3−0),(5−3),(6−2)(3−0),(5−3),(6−2)

On simplifying we get,

(3,2,4)(3,2,4)

For the linesABandCDto be perpendicular

a1a2+b1b2+c1c2=0

On substituting fora1,b1,c1a1,b1,c1anda2,b2,c2a2,b2,c2we get,

=(2×3)+(5×2)+(−4×4)=(2×3)+(5×2)+(−4×4)

=6+10−16=6+10−16

=0=0

HenceAB⊥CD