Find the vector equation of the line through the origin which is perpendicular to the plane ` vec rdot(( hat i+2 hat j+3 hat k)=3.`

Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane vec r.( hat i+2 hat j-5 hat k)+9=0 .

Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane -> r(dot( hat i+2 hat j-5 hat k))=9.

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

Find the vector equation of the line passing through the point (1,2,3) and parallel to the planes vec rdot( hat i- hat j+2 hat k)=5 and vec rdot(3 hat i+ hat j+ hat k)=6.

Write the vector equation of the line passing through (1,2,3) and perpendicular to the plane vec r.( hat i+2 hat j-5 hat k)+9=0.

Write the vector equation of the line passing through the point (1, -2, -3) and normal to the plane vec rdot((2 hat i+ hat j+2 hat k)=5.

Find the vector equation of the line passing through (1, 2, 3 ) and parallel to the planes vec rdot( (hat i- hat j+2 hat k)=5\ a n d\ vec rdot((3 hat i+ hat j+ hat k)=6.

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.

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

Find the vector and Cartesian equations of the line through the point (1, 2, 4) and perpendicular to the two lines vec r=(8 hat i-19 hat j+10 hat k)+lambda(3 hat i-16 hat j+7 hat k) and vec r=(15 hat i+29 hat j+5 hat k)+mu(3 hat i+8 hat j-5 hat k)dot