Find the value of `lambda` for which the vectors 3i-2j-k,2i+3j-4k,-i+j+2k and `4i+5j+lambdak` are coplanar.

The value of lambda for which the vectors veca= hati+ 3 hat j- hat k and vec b= 2hati+6 hat j+lambdak are parallel is

The number of distinct real values of lambda , for which the vectors -lambda^2 hat i+ hat j+k , hat i-lambda^2 hat j+ hat ka n d hat i+ hat j-lambda^2 hat k are coplanar is a. zero b. one c. two d. three

(i) Determine the value of p for which the vectors phat(i)-5hat(j) " and " 2hat(i)-3hat(j) are collinear. (ii) The sides AB and BC of the triangle ABC are represented by the vectors 2hat(i)-hat(j)+2hat(k) " and " hat(i)+3hat(j)+5hat(k) respectively, find the vector representing side CA of the triangle ABC.

Find the value of lambda for which the three points with position vector 3 hat i-2 hat j- hat k ,2 hat i+3 hat j-4 hat k , and4 hat i+5 hat j+lambda hat k are coplanar.

Determine the values of p and q for which the vectors phat(i)+2hat(j)+6hat(k) " and " 3hat(i)-3hat(j)+qhat(k) are parallel.

Find the angle between the vectors hat i-2 hat j+3 hat k and 3 hat i-2 hat j+ hat kdot

Find a unit vector perpendicular to the plane of ABC where the position vectors of A,B,C are 3i-j+2k,i-j-3k and 4i-3j+k respectively.

Find the value of lamda if three vectors vec(a) = 2hat(i) - hat(j) + hat(k), vec(b) = hat(i) + 2hat(j) - 3hat(k) and vec(c) = 3hat(i) + lamda hat(j) + 5hat(k) are coplanar

Column I, Column II The possible value of vec a if vec r=( hat i+ hat j)+lambda( hat i+2 hat i- hat k) and vec r=( hat i+2 hat j)+mu(- hat i+ hat j+a hat k) are not consistent, where lambdaa n dmu are scalars, is, p. -4 The angel between vectors vec a=lambda hat i-3 hat j- hat ka n d vec b=2lambda hat i+lambda hat j- hat k is acute, whereas vecrtor vec b makes an obtuse angel with the axes of coordinates. Then lambda may be, q. -2 The possible value of a such that 2 hat i- hat j+ hat k , hat i+2 hat j+(1+a)k a n d3 hat i+a hat j+5 hat k are coplanar is, r. 2 If vec A=2 hat i+lambda hat j+3 hat k , vec B=2 hat i+lambda hat j+ hat k , vec C=3 hat i+ hat ja n d vec A+lambda vec B is perpendicular to vec C then |2lambda| is, s. 3

The dot products of a vector with the vectors hat (i) - 3 hat(k) , hat (i) - 2hat (k) and hat (i) + hat (j)+ 4 hat (j) are 0,5,8 respectively . Find the vector .