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

Find the equation of the plane passing through (3,4,-1), which is parallel to the plane vec rdot(2 hat i-3 hat j+5 hat k)+7=0.

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

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

Find the point where line which passes through point (1,2,3) and is parallel to line vec r= hat i+ hat j+2 hat k+lambda( hat i-2 hat j+3 hat k) meets the xy-plane.

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 angle between the vectors vec a= hat i+hat j-hat k and vec b=hat i-hat j+hat k

Find vec adot vec b when: vec a= hat j- hat k\ a n d\ vec b=2 hat i+3 hat j-2 hat k

A unit vector parallel to the intersection of the planes vec r. \ ( hat i- hat j+ hat k)=5 \ a n d \ vec r. \ (2 hat i+ hat j-3 hat k)=4 a. (2 hat i+5 hat j-3 hat k)/(sqrt(38)) b. (-2 hat i+5 hat j-3 hat k)/(sqrt(38)) c. (2 hat i+5 hat j-3 hat k)/(sqrt(38)) d. (-2 hat i-5 hat j-3 hat k)/(sqrt(38))

Find the equation of a plane through the intersection of the planes vec(r). (hat(i) + 3hat(j) - hat(k)) =5 and vec(r). (2hat(i) - hat(j) + hat(k)) and passing through the point (2, 1, -2)

Find a vector magnitude 5 units, and parallel to the resultant of the vectors vec a=2 hat i+3 hat j- hat k and vec b= hat i-2 hat j+ hat kdot