If `r=hat(i)+hat(j)+lambda(2hat(i)+hat(j)+4hat(k)) and r*(hat(i)+2hat(j)-hat(k)=3` are equations of a line and a plane respectively, then which of the following is incorrect?

If [[hat(i)+4hat(j)+6hat(k), 2hat(i)+ahat(j)+3hat(k), hat(i)+2hat(j)-3hat(k)]]=0 then a=

If vector hat(i)+hat(j)+hat(k), hat(i)-hat(j)+hat(k) and 2hat(i)+3hat(j)+lambda hat(k) are coplanar, then lambda is equal to

Show that the vectors hat(i)-3hat(j)+4hat(k),2hat(i)-hat(j)+2hat(k)and 4hat(i)-7hat(j)+10hat(k) are coplanar.

Find the angle between the line: vec(r) = (hat(i) - hat(j) + hat(k) ) + lambda (2 hat(i) - hat(j) + 3 hat(k)) and the plane vec(r). (2 hat(i) + hat(j) - hat(k) ) = 4. Also, find the whether the line is parallel to the plane or not.

If a=2hat(i)+3hat(j)-hat(k), b=-hat(i)+2hat(j)-4hat(k), c=hat(i)+hat(j)+hat(k) , then find the value of (atimesb)*(atimesc) .

Find 'lambda' if the vectors : hat(i)-hat(j)+hat(k), 3hat(i)+hat(j)+2hat(k) and hat(i)+lambda hat(j)-3hat(k) are coplanar.

Show that the lines vec( r) =(hat(i) +hat(j) -hat(k)) +lambda (3hat(i) -hat(j)) " and " vec( r) =(4hat(i) -hat(k)) + mu (2hat(i) +3hat(k)) intersect . Find the point of the intersection.