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 unit 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.

If vec a.hat i= vec a.( hat i+ hat j)= vec a.( hat i+ hat j+ hat k) , then find the unit vector vec a

If points hat i+ hat j , hat i- hat j and p hat i+q hat j+r hat k are collinear, then

The dot products of a vector with the vectors hat (i) - 3 hat(k) , hat (i) - 2hat (k) and hat (i) + hat (j)+ 4 hat (j) are 0,5,8 respectively . Find the vector .

If two side of a triangle are hat i+2 hat j and hat i+ hat k , then find the length of the third side.

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

Let A ,B ,C be points with position vectors 2 hat i- hat j+ hat k , hat i+2 hat j+3 hat ka n d3 hat i+ hat j+2 hat k respectively. Find the shortest distance between point B and plane O A Cdot

The position vectors of the points A,B, and C are 2hat(i)+4hat(j)-hat(k), 4hat(i)+5hat(j)+hat(k) " and " 3hat(i)+6hat(j)-3hat(k) respectively. Show that the points form a right-angled triangle.

The position vectors of the point A ,B ,C and D are 3 hat i-2 hat j- hat k ,2 hat i+3 hat j-4 hat k ,- hat i+ hat j+2 hat k and 4 hat i+5 hat j+lambda hat k , respectively. If the points A ,B ,C and D lie on a plane, find the value of lambda .

The position vectors of the points A, B and C are 2hat(i)+6hat(j)-hat(k), hat(i)+2hat(j)+4hat(k) " and " 3hat(i)+10hat(j)-6hat(k) respectively. Show that the points A, B and C are collinear.

Let the position vectors of the points Pa n dQ be 4 hat i+ hat j+lambda hat ka n d2 hat i- hat j+lambda hat k , respectively. Vector hat i- hat j+6 hat k is perpendicular to the plane containing the origin and the points Pa n dQ . Then lambda equals a -1//2 b. 1//2 c. 1 d. none of these