Find the equations of the line through the point ( 1 , -2, -3) and parallel to the vector `2 hat i + 3 hat j + 4 hat k ` in Cartesian form.

Find the equation of the line through the point (1,2,0) and is parallel to the vector 2hat(i)+3hat(j)-5hat(k) .

Find the vector and cartesion equation of the line through the point (5,2,-4) and parallel to vector 3hat(i)+2hat(j)=8hat(k) .

Find the eqution of a line which is passes through the point (1,1,1) and parallel to 2hat(i)+hat(j)-3hat(k) .

Find the vector equation of plane that passes through the point (1,0,-2) and the normal to the plane is hat(i)+hat(j)-hat(k) .

Find the equation of the straight line passing through the point (2,-1,3) and perpendicular to the lines vec(r)=(hat(i)+hat(j)-hat(k))+lamda(2hat(i)+hat(j)-3hat(k)) and vec(r)=(hat(i)-hat(j)-hat(k))+mu(hat(i)+hat(j)+hat(k)) .

The vector equation of the plane through the point 2hat(i)-hat(j)-4hat(k) and parallel to the plane rcdot(4hat(i)-12hat(j)-3hat(k))-7=0 is

Find the projection of the vector vec a = 2 hat i+ 3 hat j+ 2 hat k on the vector vec b= hat i+ 2 hat j+ hat k .