Find the vector equation of a line passing through a point with position vector `2 hat i- hat j+ hat k ,` and parallel to the line joining the points `- hat i+4 hat j+ hat ka n d hat i+2 hat j+2 hat kdot` Also, find the Cartesian equivalent of this equation.

Find the vector equation of a line passing through the point with position vector hat i-2hat j-3hat k and parallel to the line joining the points with position vectors hat i-hat j+4hat k and 2hat i+hat j+2hat k. Also,find the cartesian equivalent of this equation.

Find the vector equation of a line passing through the point with position vector 2hat(i) + hat(j) - hat(k) and parallel to the line joining the points - hat(i) + hat(j) + 4 hat(k) and hat(i) + 2 hat(j) + 2 hat(k) .

(i) Find the vector equation of a line passing through a point with position vector 2 hat(i) - hat(j) + hat(k) and parallel to the line joining the points with position vectors - hat(i) + 4 hat(j) + hat(k) and hat(i) + 2 hat(j) + 2 hat(k). Also, find the cartesian equivalent of the equation. (ii) Find the vector equation of a line passing through the point with position vector hat(i) - 2 hat(j) - 3 hat(k) and parallel to the line joining the points with position vectors hat(i) - hat(j) + 4 hat(k) and 2 hat(i) + hat(j) + 2 hat(k) . Also, find the cartesian form of the equation.

Find the vector equation of a plane passing through a point having position vector 2 hat i- hat j+ hat k and perpendicular to the vector 4 hat i+2 hat j-3 hat kdot

Find the vector equation of a plane passing through the point having position vector 2 hat(i) + hat(j) + hat(k) and perpendicular to the vector : 4 hat(i) - 2 hat(j) + 3hat(k) .

Find the vector equation of a line passing through the point having the position vector (hat(i)+2hat(j) -3hat(k)) and parallel to the line joining the points with position vectors (hat(i) -hat(j) +5hat(k)) and (2hat(i) +3hat(j) -4hat(k)) .Also find the Cartesian equivalents of this equations.

Find the vector equation of the plane passing through the points having position vectors hat i+hat j-2hat k,2i-hat j+hat k and hat i+2hat j+hat k

Find the vector equations of the line passing through the point with position vector (2hat(i) +hat(j) -5hat(k)) and parallel to the vector (hat(i) +3hat(j) -hat(k)) .Deduce the Cartesian equations of the line .

Find the vector equation of a line which passes through the point with position vector 2hat i=hat j+4hat k and is in the direction of hat i+hat j-2hat k .Also,reduce it to Cartesian form.

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