Prove that `hat(i)times(vec atimeshat(i))+hat(j)times(vec atimeshat(j))+hat(k)times(vec atimeshat(k))=2 vec a`

Prove that ( vec a.hat i)( vec axx hat i)+( vec a.j)( vec axx hat j)+( vec a. hat k)( vec axx hat k)=0.

Prove that ( vec a.( vec bxx hat i)) hat i+( vec a.( vec bxx hat j)) hat j+( vec a.( vec bxx hat k)) hat k= vec axx vec b .

If vec(a) = hat(i) - hat(j) + 2hat(k) , vec(b) = hat(i) +hat(j) +hat(k) and vec(c ) = 2 hat(i) - hat(j) + hat(k) find the vector vec (r ) satisfying the relations , vec(a) . vec(r ) = 1 , vec(b). vec(r ) = 2 and vec(c ) . vec (r ) = 5

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 -

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

Determine whether the following pair of lines intersect or not. (1) vec r= hat i-5 hat j+lambda(2 hat i+ hat k); vec r=2 hat i- hat j+mu( hat i+ hat j- hat k) (2) vec r= hat i+ hat j- hat k+lambda(3 hat i- hat j); vec r=4 hat i- hat k+mu(2 hat i+3 hat k)

The lines vec(r)=hat(i)+t(5hat(i)+2hat(j)+hat(k)) and vec(r)=hat(i)+s(-10hat(i)-4hat(j)-2hat(k)) are -

If vec a.hat i= vec a.( hat i+ hat j)= vec a.( hat i+ hat j+ hat k) , then find the unit vector vec a

If vec(a)=hat(i)+hat(j)+hat(k), vec(b)=2hat(i)-hat(j)+3hat(k) " and " vec(c)=hat(i)-2hat(j)+hat(k) find a unit vector in a direction parallel to vector (2vec(a)-vec(b)+3vec(c)) .

Consider the vectors vec(A)=hat(i)+hat(j)-hat(k),vec(B)=2hat(i)-hat(j)+hat(k),vec(C)=(1)/(sqrt(5))(hat(i)-2hat(j)+2hat(k)) . What is the value of vec(C).(vec(A)xxvec(B)) ?