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 -> r(dot( hat i+2 hat j-5 hat k))=9.

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 perpendicular to the plane vec r*(hat i+2hat j-5hat k)+9=0

Find the unit vector perpendicular to the plane vec r*(2hat i+hat j+2hat 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.

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

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

Find the equation of the line passing through the point hat i+ hat j- hat k and perpendicular to the lines vec r= hat i+lambda(2 hat i+ hat j-3 hat k)a n d vec r=(2 hat i+ hat j- hat k)dot

Find the vector and Cartesian equations 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 equation of the plane through the line of intersection of the planes vec rdot(( hat i+3 hat j)+6=0\ a n d\ vec rdot((3 hat i- hat j-4 hat k)=0, which is at a unit distance from the origin.