[" The vectors "],[vec u=(a|+a lambda_(1))i+(am+a,m,)j+(an+an,j)dot k],[vec v=(bl+bb_(1))i+(bm+bm_(i))j+(bn+b,n,j)],[bar(w)=(d+c,b)i+(cm+cm_(4))j+(cn+c,n_(i))dot k]

Prove that vectors vec u=(al+a_(1)l_(1))hat i+(am+a_(1)m_(1))hat j+(an+a_(1)n_(1))hat kvec v=(bl+b_(1)l_(1))hat i+(bm+b_(1)m_(1))hat j+(bn+b_(1)n_(1))hat kvec w=(bl+b_(1)l_(1))hat i+(bm+b_(1)m_(1))hat j+(bn+b_(1)n_(1))hat k are coplanar.

Find the value of lambda for which the vectors vec(a), vec(b), vec(c) are coplanar, where (i) vec(a)=(2hat(i)-hat(j)+hat(k)), vec(b) = (hat(i)+2hat(j)+3hat(k) ) and vec(c)=(3 hat(i)+lambda hat(j) + 5 hat (k)) (ii) vec(a)lambda hat(i)-10 hat(j)-5k^(2), vec(b) =-7hat(i)-5hat(j) and vec(c)= hat(i)--4hat(j)-3hat(k) (iii) vec(a)=hat(i)-hat(j)+hat(k), vec(b)= 2hat( i) + hat(j)-hat(k) and vec(c)= lambda hat(i) - hat(j) + lambda hat(k)

Determine lambda such that a vector vec(r) is at right angles to each of the vectors vec(a) = lambda hat(i) + hat(j) + 3hat(k), vec(b) = 2hat(i) + hat(j) - lambda hat(k), vec(c) = -2hat(i) + lambda hat(j) + 3hat(k) .

vectors vec a=i-j+2k,vec b=2i+4j+k and vec c=lambda i+j+mu k are mutually orthogonal then (lambda,mu) is

Find the value of lambda such that the vectors vec a = 2i + lambda j+ k, vec b = i + 2j +3k are orthogonal

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

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

If the vectors vec A = a hat i + hat j + hat k, vec B = hat i + b hat j + hat k and vec C = hat i + hat j + chat k are coplanar, then (1)/(1-a) + (1)/(1-b) +(1)/(1-c) = 1 where a,b, c ne 1 .

If vec a = i+j+k, vec a. vec b = 1 and vec a xx vec b = j-k then vec b =

If the position vectors of A and B are 3i - 2j + k and 2i + 4j - 3k then |vec(A)B|