The position vectors of the point `A ,B ,Ca n dDa r e3 hat i-2 hat j- hat k ,2 hat i+3 hat j-4 hat k ,- hat i+ hat j+2 hat k` and 4` hat i+5 hat j+lambda hat k ,` respectively. If the points `A ,B ,Ca n dD` lie on a plane, find the value of `lambdadot`

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

Prove that point hat i+2hat j-3hat k,2hat i-hat j+hat k and 2hat i+5hat j-hat k from a triangle in space.

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

Show that the points A(-2 hat i+3 hat j+5 hat k), B( hat i+2 hat j+3 hat k) and C(7 hat i-3 hat k) are collinear.

Show that the point A , B and C with position vectors vec a =3 hat i - 4 hat j -4 hat k = 2 hat i j + hat k and vec c = hat i - 3 hat j - 5 hat k , respectively from the vertices of a right angled triangle.

The position vectors of three points A,B anc C (-4hat(i) +2hat(j) -3hat(k)) (hat(i) +3hat(j) -2hat(k)) "and " (-9hat(i) +hat(j) -4hat(k)) respectively . Show that the points A,B and C are collinear.

The position vectors of the points Pa n dQ with respect to the origin O are vec a= hat i+3 hat j-2 hat k and vec b=3 hat i- hat j-2 hat k , respectively. If M is a point on P Q , such that O M is the bisector of angleP O Q , then vec O M is a. 2( hat i- hat j+ hat k) b. 2 hat i+ hat j-2 hat k c. 2(- hat i+ hat j- hat k) d. 2( hat i+ hat j+ hat k)

If vec r=(hat i+2hat j+3hat k)+lambda(hat i-hat j+hat k) and vec r=(hat i+2hat j+3hat k)+mu(hat i+hat j-hat k) are bisector of two lines is

Find the vector of length 3 unit which is perpendicular to hat i+hat j+hat k and lies in the plane of hat i+hat j+hat k and 2hat k-3hat j

Equation of the plane containing the lines vec r=(hat i-2hat j+hat k)+t(hat i+2hat j-hat k)vec r=(hat i+2hat j-hat k)+s(hat i+hat j+3hat k) is