The position vectors of the vertices `A ,B ,a n dC` of a triangle are ` hat i+ hat j , hat j+ hat ka n d hat i+ hat k` , respectively. Find the unite vector ` hat r` lying in the plane of `A B C` and perpendicular to `I A ,w h e r eI` is the incentre of the triangle.

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

If the position vector of the vertices of a triangle are hat(i)-hat(j)+2hat(k), 2hat(i)+hat(j)+hat(k) & 3hat(i)-hat(j)+2hat(k) , then find the area of the triangle.

Dot products of a vector with vectors hat i- hat j+ hat k ,\ 2 hat i+ hat j-3 hat k\ a n d\ hat i+ hat j+ hat k are respectively 4, 0 and 2. Find the vector.

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.

If the position vectors of the points A , B , C ,Da r e2 hat i+4 hat k ,5 hat i+3sqrt(3) hat j+4 hat k ,-2sqrt(3) hat j+ hat ka n d2 hat i+ hat k respectively, prove that C D is parallel to A Ba n dC D=2/3A Bdot

Dot product of a vector with hat i+ hat j-3 hat k ,\ hat i+3 hat j-2 hat k\ a n d\ 2 hat i+ hat j+4 hat k are 0, 5 and 8 respectively. Find the vector.

The position vectors of point A ,B ,a n dC are hat i+ hat j+ hat k , hat i+5 hat j- hat ka n d2 hat i+3 hat j+5 hat k , respectively. Then greatest angel of triangle A B C is 120^0 b. 90^0 c. cos^(-1)(3//4) d. none of these

The position vectors of the points A ,\ B ,\ C are 2 hat i+ hat j- hat k ,\ 3 hat i-2 hat j+ hat k\ a n d\ hat i+4 hat j- hat k respectively. These points a.Form an isosceles triangle b.Form a right triangle c.Are collinear d.Form a scalene triangle

If the position vectors of the point A and B are 3hat(i)+hat(j)+2hat(k) and hat(i)-2hat(j)-4hat(k) respectively. Then the eqaution of the plane through B and perpendicular to AB is

Find a vector of magnitude 9, which is perpendicular to both vectors 4 hat i- hat j+3 hat ka n d-2 hat i+ hat j-2 hat k .