Reduce the equation ` vec r (dot(3 hat i-4 hat j+12 hat k ))=5` to normal form and hence find the length of perpendicular from the origin to the plane.

Write the plane vec rdot((2 hat i+3 hat j-6 hat k)=14 in normal form.

Write the the projections of vec r=3 hat i-4 hat j+12 hat k on the coordinate axes.

Find the equation of the plane through the line of intersection of vec r dot(2 hat i-3 hat j+4 hat k)=1a n d vec r dot( hat i- hat j)+4=0 and perpendicular to vec r dot(2 hat i- hat j+ hat k)+8=0.

Find the equation the plane which contain the line of intersection of the planes vec rdot( hat i+2 hat j+3 hat k)-4=0a n d vec rdot(2 hat i+ hat j- hat k)+5=0 and which is perpendicular to the plane vec r(5 hat i+3 hat j-6 hat k)+8=0 .

Find the vector equation of the plane which contains the line of intersection of the plane vec rdot( hat i+2 hat j+3 hat k)-\ 4=0 and vec rdot(2 hat i+ hat j- hat k)+5=0 and which is perpendicular to the plane vec rdot(5 hat i+3 hat j-\ 6 hat k)+8=0.

Find the direction cosines of perpendicular from the origin to the plane vec rdot(2 hat i-3 hat j-6 hat k)+5=0.

Find the unit vector perpendicular to the plane vec rdot(2 hat i+ hat j+2 hat k)=5.

Find the equation of the plane which contains the line of intersection of the planes vec r . ( hat i+2 hat j+3 hat k)-4=0,\ vec r . (2 hat i+ hat j- hat k)+5=0 and which is perpendicular to the plane vec r . (5 hat i+3 hat j-6 hat k)+8=0.

Find a vector of magnitude 49, which is perpendicular to both the vectors 2 hat i+3 hat j+6 hat k and 3 hat i-6 hat j+2 hat k . Find a vector whose length is 3 and which is perpendicular to the vector vec a=3 hat i+ hat j-4 hat k and vec b=6 hat i+5 hat j-2 hat k .

Find the coordinates of the foot of the perpendicular and the length of the perpendicular drawn from the point P(5,4,2) to the line vec(r) = -hat(i) + 3hat(j) + hat(k) + lambda (2hat(i) + 3hat(j) - hat(k)) .