[" 41.A point on the line "],[qquad bar(r)=(1-t)(2i+3bar(j)+4bar(k))+t(3bar(i)-2bar(j)+2bar(k))]

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

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

Compute 2 bar(j) xx (3bar(i) - 4bar(k)) + (bar(i)+2bar(j)) xx bar(k) .

Find the shortest distance between the skew lines . bar(r)=(6bar(i)+2bar(j)+2bar(k))+t(bar(i)-2bar(j)+2bar(k))andbar(r)=(-4bar(i)-bar(k))+s(3bar(i)-2bar(j)-2bar(k))

Find the vector equation of the line passing through the points 2bar(i)+3bar(j)+4bar(k), 3bar(i)-2bar(j)+2bar(k) .

The point of intersection of the line passing through bar(i)-2bar(j)-bar(k), 2bar(i)+3bar(j)+bar(k) and the plane passing through 2bar(i)+bar(j)-3bar(k), 4bar(i)-bar(j)+2bar(k), 3bar(i)+bar(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

If the position vectors of the points A,B,C are -2bar(i)+bar(j)-bar(k), -4bar(i)+2bar(j)+2bar(k), 6bar(i)-3bar(j)-13bar(k) respectively and bar(AB) = lambda bar(AC) then find the value of lambda .

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 vectors 2bar(i)-3bar(j)+bar(k), bar(i)-2bar(j)+3bar(k), 3bar(i)+bar(j)-2bar(k)