Find `vec(A).vec(b)` when
(i) `vec(a)=hat(i)-2hat(j)+hat(k)` and `vec(b)=3 hat(i)-4 hat(j)-2 hat(k)`
(ii) `vec(a)=hat(i)+2hat(j)+3hat(k)` and `vec(b)=-2hat(j)+4hat(k)`
(iii) `vec(a)=hat(i)-hat(j)+5hat(k)` and `vec(b)=3 hat(i)-2 hat(k)`

Find the angle between the vectors vec(a) and vec(b) , when (i) vec(a)=hat(i)-2hat(j)+3 hat(k) and vec(b)=3hat(i)-2hat(j)+hat(k) (ii) vec(a)=3 hat(i)+hat(j)+2hat(k) and vec(b)=2hat(i)-2hat(j)+4 hat(k) (iii) vec(a)=hat(i)-hat(j) and vec(b)=hat(j)+hat(k) .

Find ( vec (a) xxvec (b)) and |vec(a) xx vec (b)| ,when (i) vec(a) = hat(i)-hat(j)+ 2hat(k) and vec(b)= 2 hat(i)+3 hat(j)-4hat(k) (ii) vec(a)= 2hat (i)+hat(j)+ 3hat(k) and vec(b)= 3hat(i)+5 hat(j) - 2 hat(k) (iii) vec(a)=hat(i)- 7 hat(j)+ 7hat(k) and vec(b) = 3 hat(i)-2hat(j)+2 hat(k) (iv) vec(a)= 4hat(i)+ hat(j)- 2hat(k) and vec(b) = 3 hat(i)+hat(k) (v) vec(a) = 3 hat(i) + 4 hat(j) and vec(b) = hat(i)+hat(j)+hat(k)

Find the area of the parallelogram whose adjacent sides are represented by the vectors (i) vec(a)=hat(i) + 2 hat(j)+ 3 hat(k) and vec(b)=-3 hat(i)- 2 hat(j) + hat(k) (ii) vec(a)=(3 hat(i)+hat(j) + 4 hat(k)) and vec(b)= ( hat(i)- hat(j) + hat(k)) (iii) vec(a) = 2 hat(i)+ hat(j) +3 hat(k) and vec(b)= hat(i)-hat(j) (iv) vec(b)= 2 hat(i) and vec(b) = 3 hat(j).

Find the value of lambda for which vec(a) and vec(b) are perpendicular, where (i) vec(a)=2hat(i)+lambda hat(j)+hat(k) and vec(b)=(hat(i)-2hat(j)+3hat(k)) (ii) vec(a)=3hat(i)-hat(j)+4hat(k) and vec(b)=- lamnda hat(i)+3 hat(j)+3 hat(k) (iii) vec(A)=2hat(i)+4hat(j)-hat(k) and vec(b)=3 hat(i)-2 hat(j)+lambda hat(k) (iv) vec(a)=3 hat(i)+2 hat(j)-5 hat(k) and vec(b)=-5 hat(j)+lambda hat(k)

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

verify that vec(a) xx (vec(b)+ vec(c))=(vec(a) xx vec(b))+(vec(a) xx vec(c)) , "when" (i) vec(a)= hat(i)- hat(j)-3 hat(k), vec(b)= 4 hat(i)-3 hat(j) + hat(k) and vec(c)= 2 hat(i) - hat(j) + 2 hat(k) (ii) vec(a)= 4 hat(i)-hat(j)+hat(k), vec(b)= hat(i)+hat(j)+ hat(k) and vec(c)= hat(i)- hat(j)+hat(k).

Find [vec(a)vec(b)vec(c)] , when (i) vec(a)=2hat(i)+hat(j)+3hat(k), vec(b)=-hat(i)+2hat(j)+hat(k) and vec(c)=3hat(i)+hat(j)+2hat(k) (ii) vec(a)=2hat(i)-3hat(j)+4hat(k), vec(b)=hat(i)+2hat(j)-hat(k) and vec(c)=3hat(i)-hat(j)+2hat(k) (iii) vec(a) = 2 hat(i)-3hat(j), vec(b)=hat(i)+hat(j)-hat(k) and vec(c)=3hat(i)-hat(k)