The position vectors of the points A,B,C,D are `vec(3i)-vec(2j)-veck, vec(2i)+vec(3j)-vec(4k)-veci+vecj+vec(2k) and vec(4j) +vec(5j)+vec(lamdak)` respectively Find `lamda` if A,B,C,D are coplanar.

The vector area of the triangle formed by the points vec(i) -vec(j) + vec(k), 2vec(i) + vec(j) -2vec(k) and 3vec(i) + vec(j) + 2vec(k) is

If vec(a)= 2vec(i)-vec(j) +vec(k), vec(b)= 3vec(i) + 4vec(j) -vec(k) , then |vec(a) xx vec(b)|=

The position vectors of the points A,B,C , D are respectively 2 vec i + vec j - vec k, vec i + vec j + vec k, vec i- 2 vec j+ 3 vec k and 3 vec i- vec j + 2 vec k . Evaluate [vec (AB), vec (AC), vec (AD)] .

The position vectors of the vertices of a triangle are vec(i) +2 vec(j) +3 vec(k) , 3 vec(i) -4 vec(j) +5 vec (k) and -2vec (i) +3 vec (j) - 7 vec (k) Find the perimeter of a triangle .

If vec(a)= 3 vec(i)- vec(j)-2vec(k), vec(b)= 2vec(i) + 3vec(j) + vec(k) , then (vec(a) + 2vec(b)) xx (2vec(a) - vec(b)) =

Examine if vec(i) - 3vec(j) + 2vec(k), 2vec(i) - 4vec(j) - vec(k) and 3vec(i) + 2vec(j) - vec(k) are linearly independent or dependent .

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

If vec(a)= vec(i) + vec(j)+ vec(k), vec(c )= vec(j)- vec(k) , then find vector vec(b) such that vec(a) xx vec(b)= vec(c ) and vec(a).vec(b)= 3

If vec(a)= 2vec(i)-3 vec(j) + vec(k). vec(b)= vec(i) + 4vec(j)- 2vec(k) , then find (vec(a) + vec(b)) xx (vec(a) - vec(b)) .

Find the vector equation of the plane passing through the points. vec(i)-2vec(j)+5vec(k), -5vec(j)-vec(k) and -3vec(j)+5 vec(j) .