Show that the four points A, B, C and D with position vectors`4 hat i+5 hat j+ hat k ,-( hat j+ hat k),3 hat j+9 hat j+4 hat k a n d 4( hat i+ hat j+ hat k)`, respectively are coplanar.

Show that the four points A, B, C and D with position vectors`4 hat i+

1.	Show that the four points A, B, C and D with position vectors`4 hat i+5 hat j+ hat k ,-( hat j+ hat k),3 hat j+9 hat j+4 hat k a n d 4( hat i+ hat j+ hat k)`, respectively are coplanar.
Answer» Let the given points be A, B, C, D respectively. Point a, B, C, D are coplanar `hArr vec(AB), vec(AC)and vec(AD)` are coplanar `hArr [vec(AB)vec(AC)vec(AD)]=0`. Now, `vec(AB)=("p.v.of B")-("p.v. of A")` `=(-hat(j)-hat(k))-(4hat(i)+5hat(j)+hat(k))=(-4hat(i)-6hat(j)-2hat(k))` `vec(AC)=("p.v of C")-("p.v. of A")` `=(3 hat(i)+9hat(j)+4hat(k))-(4hat(i)+5hat(j)+hat(k))=(-hat(i)+4hat(j)+3hat(k))` `vec(AD)=("p.v. of D")-("p.v. of A")` `=(-4hat(i)+4hat(j)+4hat(k))-(4hat(i)+5hat(j)+hat(k))=(-8hat(i)-hat(j)+3hat(k))`. `:. [vec(AB)vec(AC)vec(AD)]=\|(-4,-6,-2),(-1 ,4,3),(-8,-1,3)\|` `=\|(0,-22, -14),(-1, 4, 3),(0, -21, -33)\|{{:(R_(1)rarrR_(1)-4R_(2)),(R_(3)rarr R_(3)-8R_(2)):}}` `=-(-1)[462-462]=0`. `:. vec(AB), vec(AC) and vec(AD)` are coplanar. Hence, the points A, B, C, D are coplanar.