Find the position vector of point R which divides the line joining the points P and Q whose position vectors are `2 hat(i) - hat(j) + 3 hat(k) and -5 hat(i) + 2 hat(j) - 5 hat(k)` in the ratio 3 : 2
(i) internally
(ii) externally.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are hat i+2hat j-hat k and respectively,in the ratio 2:1(i) internally (ii) externally

Find the vector and Cartesian equations of the line joining the points whose position vectors are (hat(i) -2hat(j) +hat(k)) " and " (hat(i) +3hat(j) -2hat(k))

If the vectors 2 hat(i) - q hat(j) + 3 hat(k) and 4 hat(i) - 5 hat(j) + 6 hat(k) are collinear then find the value of q.

(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.

Show that the points whose position vectors are (5hat(i) + 6 hat(j) + 7 hat(k)), ( 7 hat(i) - 8hat(j) + 9 hat(k)) and (3 hat(i) + 20 hat(j) + 5 hat(k)) are collinear.

show that the points whose position vectors are (-2hat(i) +3hat(j) +5hat(k)) (hat(i)+2hat(j)+3hat(k)) " and " (7hat(i) -hat(k)) are collinear.

Find the distance between the points A and B with position vectors hat i-hat j and 2hat i+hat j+2hat 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.

If vector hat(i) - 3hat(j) + 5hat(k) and hat(i) - 3 hat(j) - a hat(k) are equal vectors, then the value of a is :

Write the position vector of a point dividing the line segment joining points having position vectors hat i+hat j-2hat k and 2hat i-hat j+3hat k externally in the ratio 2:3 .