Show that the vectors `vec(a), vec(b), vec(c)` are coplanar, when
(i) `vec(a)=hat(i)-2hat(j)+3hat(k), vec(b) = -2hat(i)+3hat(j)-4hat(k) and vec(c)=hat(i)-3hat(j)+5hat(k)`
(ii) `vec(a)=hat(i)+3hat(j)+hat(k), vec(b)=2hat(i)-hat(j)-hat(k)and vec(c)=7hat(j)+3hat(k)`
(iii) `vec(a)=2hat(i)-hat(j)+2hat(k), vec(b)=hat(i)+2hat(j)-3hat(k) and vec(c)=3hat(i)-4hat(j)+7hat(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)

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)

If 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) , find [hat(a)hat(b)hat(c)] .

Show that the vectors : vec(a)=hat(i)-2hat(j)+3hat(k), vec(b)=-2hat(i)+3hat(j)-4hat(k) and vec(c )=hat(i)-3hat(j)+5hat(k) are coplanar.

Find vec(a).(vec(b)xx vec(c )) if : 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) .

Vector vec(A)=hat(i)+hat(j)-2hat(k) and vec(B)=3hat(i)+3hat(j)-6hat(k) are :

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) .

If vec(a)=2hat(i)+3hat(j)+hat(k), vec(b)=hat(i)-2hat(j)+hat(k) and vec(c )=-3hat(i)+hat(j)+2hat(k) , find [vec(a)vec(b)vec(c )] .