If `alpha=2vec(i) +3vec(j) -5vec(k) and beta= vec(i)-vec(j) `, then find the value of `alpha . beta`

If vec(a) = 3 alpha hat(i) + 2hat(j) - 3 hat (k) , vec(b) = hat(i) + 6 alpha hat(j) - 2 hat(k) and hat (c ) = 2 hat(i) - 3 alpha hat(j) + hat(k) be such that {(vec(a) xx vec(b))xx (vec(b) xx vec(c)) } xx (vec(c) xx vec(a)) = vec(O) , then the value of 9 alpha is -

If vec a=2 vec i+3 vec j- vec k , vec b=- vec i+2 vec j-4 vec k a n d vec c= vec i+ vec j+ vec k , then find thevalue of ( vec axx vec b)dot(( vec axx vec c)dot)

vec (alpha) = lambda hat (i) + hat (j) + 3 hat (k) , " " vec(beta) = -hat (i) + 2 hat (j) + hat (k) ," " vec(gamma) = 3 hat (i) + hat (j) + 2 hat (k) and [ vec(alpha) vec(beta)vec(gamma)]= -10 then find the value of lambda

If vec a=a_1 hat i+a_2 hat j+a_3 hat k ; vec b=b_1 hat i+b_2 hat j+b_3 hat k , . vec c=c_1 hat i+c_2 hat j+c_3 hat k and [3 vec a+ vec b """ 3 vec b+ vec c """" 3 vec c+ vec a]=lambda[vec a vec b vec c] , then find the value of lambda/4 .

Let vec(a) = - hat(i) - hat(k) , vec(b) = - hat(i) +hat(j) and vec(c) = hat(i) + 2hat(j) + 3 hat(k) be three given vectors . If vec(r ) is a vector such that vec(r ) xx vec(b) = vec (c ) xx vec(b) and vec (r ) . vec (a ) = 0 then the value of vec(r ). vec(b) is -

Let vec(a)=4hat(i)+3hat(j)-hat(k), vec(b)=5hat(i)+2hat(j)+2hat(k), vec(c)=2hat(i)-2hat(j)-3hat(k) " and "vec(d)=4hat(i)-4hat(j)+3hat(k), " prove that " vec(b)-vec(a) " and " vec(d)-vec(c) are parallel and find the ratio of their moduli.

If vec (a) = 3 hat (i) + 2 hat (j) + 9 hat (k) and vec(b) = hat (i) + lambda hat (j) + 3 hat (k) , then find the value of lambda so that the vectors (vec(a) + vec(b)) and (vec(a) - vec(b)) are perpendicular to each other .

If vec (a) = 2 hat (i) - 2 hat (j) + hat (k) , vec(b) = hat (i) + hat (j) - hat (k) and |vec (a) xx vec(b)| = sqrt(13 m ) then the value of m is -

If vec(a)=2hat(i)-5hat(j)+3hat(k) " and " vec(b)=hat(i)-2hat(j)-4hat(k) , find the value of abs(3vec(a)+2vec(b)).

If vec(a) = hat(i) + 2 hat(j) - 3 hat (k) and vec(beta) = 3 hat(i) - hat (j) + 2 hat (k) , find the cosine of the angle between the vectors (2 vec(alpha)+ vec(beta)) and (vec(alpha)+2 vec(beta))