The distance between a point P whose position vector is `5bar(i)+bar(j)+3bar(k)` and the line `bar(r)=(3bar(i)+7bar(j)+bar(k))+t(bar(j)+bar(k))`

A point on the line bar(r)=(1-t)(2bar(i)+3bar(j)+4bar(k))+t(3bar(i)-2bar(j)+2bar(k))

The distance between the line bar(r)=2bar(i)-2bar(j)+3bar(k)+lambda((bar(i)-bar(j)+4bar(k)) and the plane bar(r).(bar(i)+5bar(j)+bar(k))=5 is

The vectors bar(i)+4bar(j)+6bar(k),2bar(i)+4bar(j)+3bar(k) and bar(i)+2bar(j)+3bar(k) form

For any vector bar(r) , (bar(r)*bar(i))bar(i)+(bar(r)*bar(j))bar(j)+(bar(r)*bar(k))bar(k)=

The shortest distance between the lines bar(r)=3bar(i)+5bar(j)+7bar(k)+lambda(bar(i)+2bar(j)+bar(k)) and bar(r)=-bar(i)-bar(j)+bar(k)+mu(7bar(i)-6bar(j)+bar(k)) is

The lines bar(r)=bar(i)+bar(j)-bar(k)+s(3bar(i)-bar(j)) and bar(r)=4bar(i)-bar(k)+t(2bar(i)+3bar(k))

The shortest distance the two lines bar(r)=2bar(i)-bar(j)-bar(k)+lambda(2bar(i)+bar(j)+2bar(k)) and bar(r)=(bar(i)+2bar(j)+bar(k))+mu(bar(i)-bar(j)+bar(k)) is

The direction ratios of a line which is perpendicular to the two vectors -2bar(i) +3 bar(j) -bar(k) and 4 bar(i) +bar(j) +3 bar(k) is

undersetbar(a)bar(a)=2bar(i)+bar(j)+3bar(k),bar(b)=pbar(i)+bar(j)+qbar(k) and bar(b)xxbar(a)=bar(0)

undersetbar(a)bar(a)=2bar(i)+bar(j)+3bar(k),bar(b)=pbar(i)+bar(j)+qbar(k) and bar(b)xxbar(a)=bar(0)