The position of vectors of the points P, Q and R are `hat p`, `hat q` and `hat r` respectively. A vector `vec v=k(hat q + hat r)` is such that `hat p` .`hat v` = `hat q. hat v`, where k is a scalar. Prove that `(hat p - hat r)`.`(hat p - hat q) = 0`

The position vectors of the points P,Q,R are hat i+2hat j+3hat k,-2hat i+3hat j+5hat k and 7hat i-hat k respectively.Prove that P,Q and R are collinear points.

The position vectors of the points P and Q are 5 hat i+7 hat j- 2 hat k and -3 hat i+3 hat j+6 hat k respecively. The vector vec A=3 hat i-hat j+hat k passes through the point P and the vector vec B=-3 hat i+2 hat j+4 hat k passes through the point Q. A third vector 2 hat i+7 hat j- 5hat k intersects vectors vec A and vec B . Find the position vectors of the points of intersection

The position vector of the points Pa n dQ are 5 hat i+7 hat j-2 hat k and -3 hat i+3 hat j+6 hat k , respectively. Vector vec A=3 hat i- hat j+ hat k passes through point P and vector vec B=-3 hat i+2 hat j+4 hat k passes through point Q . A third vector 2 hat i+7 hat j-5 hat k intersects vectors Aa n dBdot Find the position vectors of points of intersection.

The position vectors of P and Q are 5hat i+4hat j+ahat k and -hat i+2hat j-2hat k, respectively. If the distance between them is 7, then find the value of a.

The position vector of a point is R =x hat i +y hat j + z hat k and another vector is A =3 hat i +2 hat j + 5hat k .Which of the mathematical relations is correct ?

The position vectors of points AandB w.r.t. the origin are vec a=hat i+3hat j-2hat k ,vec b=3hat i+hat j-2hat k respectively.Determine vector vec OP which bisects angle AOB, where P is a point on AB.

vec A B=3 hat i- hat j+ hat ka n d vec C D=-3 hat i+2 hat j+4 hat k are two vectors. The position vectors of the points Aa n dC are =6 hat i+7 hat j+4 hat ka n d=-9 hat j+2 hat k respectively. Find the position vector of a point P on the line A B and a point Q on the line C D such that vec P Q is perpendicular to vec A Ba n d vec C B both.

For any vector vec r ,prove that vec r=(vec r*hat i)hat i+(vec r*hat j)hat j+(vec r*hat k)hat k

vec [AB]=3hat i+ hat j+ hat k and vec [CD]= -3hat i+2 hat j+4 hat k are two vectors. The position vectors of points A and C are 6 hat i+ 7 hat j+ 4 hat k and -9 hat j +2 hat k respectively. Find the position vector of point P on the line AB and point Q on the line CD such that vec [PQ] is perpendicular to vec [AB] and vec [CD] both.

The vectors vec P = a hat I + a hat j + 3 hat k) and vec Q = a hat I - 2 hat j - hat kl are peroendicular to each other. The positve value of (a) is.