Find the mid point of the line segment joining the points `P(2hati+3hatj+3hatk) and Q(4hati+hatj-2hatk)`

If M denotes the md point of the line segment joining A(4hati+6hatj-10hatk) and B(-hati+2hatj+hatk) , then the equation, of the plane through M and perpendicular to AB is

Find the position vector of a point R which divides the line joining the point P(hati + 2hatj - hatk) and Q(-hati + hatj + hatk) in the ratio 2 : 1 , (i) internally and (ii) externally.

Find the position vector of a point R which divides the line joining the point P(hati + 2hatj - hatk) and Q(-hati + hatj + hatk) in the ratio 2 : 1 internally .

The ratio in which the plane vecr.( hati-2 hatj+3 hatk)=17 divides the line joining the points -2 hati+4 hatj+7 hatk and 3 hati-5 hatj+8 hatk Is __

Find the points of intersection of the line vecr = (2hati - hatj + 2hatk) + lambda(3hati + 4hatj + 2hatk) and the plane vecr.(hati - hatj + hatk) = 5

The position vector of the point in which the line joining the points hati-2hatj+hatk and 3hatk-2hatj cuts the plane through the origin and the points 4hatj and 2hati+hatk is

Find the angle between the vectors 4hati-2hatj+4hatk and 3hati-6hatj-2hatk.

Find the vector equation of a line passing thorugh a point with position vector 2hati-2hatj+hatk and parallel to the line joining the point with the position vectors -hati+3hatj+hatk and 2hati+hatj+3hatk . Also find the Cartesian equation of this line.

The vector equation of the plane containing he line vecr=(-2hati-3hatj+4hatk)+lamda(3hati-2hatj-hatk) and the point hati+2hatj+3hatk is

Find the equation of the plane through the point hati+4hatj-2hatk and perpendicular to the line of intersection of the planes vecr.(hati+hatj+hatk)=10 and vecr.(2hati-hatj+3hatk)=18.