Find `bar(a). bar(b) xx bar(c)`, if `bar(a) = 3 hat(i) - hat(j) + 4 hat(k), bar(b) = 2hat(i) + 3 hat(j) - hat(k), bar(c) = - 5 hat(i) + 2 hat(j) + 3 hat(k)`

If bar(a) = 4 hat(i) + 3 hat(k) and bar(b) = - 2 hat(i) + hat(j) + 5 hat(k) then find 2 bar(a) + 5 bar(b)

Find [bar(a)bar(b)bar(c)] where : (1) bar(a)=2hat(i)+hat(j)-hat(k),bar(b)=3hat(i)-hat(j)-hat(k),hat(c)=hat(j)+3hat(k) (2) bar(a)=7hat(i)-hat(j)+2hat(k),bar(b)=hat(i)+3hat(j)-hat(k),bar(c)=4hat(i)+5hat(k) .

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 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).

Express bar(p) as a linear combination of bar(a),bar(b)andbar(c) , where bar(p)=hat(i)+4hat(j)-4hat(k),bar(a)=2hat(i)-hat(j)+3hat(k),bar(b)=hat(i)-2hat(j)+4hat(k),bar(c)=-hat(i)+3hat(j)-5hat(k) .

If bar(a)=3hat(i)-2hat(j)+7hat(k),bar(b)=5hat(i)+hat(j)-2hat(k)andbar(c)=hat(i)+hat(j)-hat(k) , then find bar(a)*(bar(b)xxbar(c)) .

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)

If bar(a)=3hat(i)-hat(j)+4hat(k)," "bar(b)=2hat(i)+3hat(j)-hat(k)," "bar(c)=-5hat(i)+2hat(j)+3hat(k) , then bar(a)*(bar(b)xxbar(c)) = . . . . . . . .

Find lamda , if the vectors bar(a)=hat(i)+hat(j)+hat(k),bar(b)=hat(i)-hat(j)+hat(k)andbar(c)=2hat(i)+3hat(j)+lamdahat(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)