The reciprocal of `bar(a)` where `bar(a)=-bar(i)+bar(j)+bar(k),bar(b)=bar(i)-bar(j)+bar(k),bar(c)=bar(i)+bar(j)+bar(k)` is

undersetbar(a)bar(a)=2bar(i)+bar(j)+3bar(k),bar(b)=pbar(i)+bar(j)+qbar(k) and bar(b)xxbar(a)=bar(0)

undersetbar(a)bar(a)=2bar(i)+bar(j)+3bar(k),bar(b)=pbar(i)+bar(j)+qbar(k) and bar(b)xxbar(a)=bar(0)

Consider the parallelopiped with sides bar(a)=3bar(i)+2bar(j)+bar(k),bar(b)=bar(i)+bar(j)+2bar(k) and bar(c)=bar(i)+3bar(j)+3bar(k) then angle between bar(a) and the plane containing the face determined by bar(b) and bar(c) is

If three vectors bar(a)=bar(i)+bar(j)+bar(k),bar(b)=bar(i)-2a^(2)bar(j)+abar(k),bar(c)=bar(i)+(a+1)bar(j)-abar(k) are linearly dependent vectors then the real a lies in the interval.

If there vectors bar(a)=bar(i)+bar(j)+bar(k) , bar(b)=bar(i)-2a^(2)bar(j)+abar(k) , bar(c)=bar(i)+(a+1)bar(j)-abar(k) are linearly dependent vectors then the real a lies in the interval.

If there vectors bar(a)=bar(i)+bar(j)+bar(k),bar(b)=bar(i)-2a^(2)bar(j)+bar(a) ,bar(c)=bar(i)+(a+1)bar(j)-abar(k) are linearly dependent vectors then the real 'a' lies in the interval.

The lines bar(r)=bar(i)+bar(j)-bar(k)+s(3bar(i)-bar(j)) and bar(r)=4bar(i)-bar(k)+t(2bar(i)+3bar(k))

bar (a) = 2bar (i) + 3bar (j) -bar (k), bar (b) = bar (i) + 2bar (j) -4bar (k), bar (c) = bar (i) + bar (j) + bar (k), bar (d) = bar (i) -bar (j) -bar (k) then

The orthogonal projection of bar(a)=2bar(i)+3bar(j)+3bar(k) on bar(b)=bar(i)-2bar(j)+bar(k) (where bar(i)*bar(j)*bar(k) are unit vectors along there mutually perpendicular directions is

bar(a)=2bar(i)+bar(j)+3bar(k),bar(b)=3bar(i)+2bar(j)+bar(k),bar(c)=bar(i)-bar(j)-4bar(k),bar(d)=bar(i)+2bar(j)-bar(k) then (bar(a)xxbar(b))xx(bar(c)xxbar(d))=