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

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

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.

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 .

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

Prove that point hat i +2 hat j - 3 hat k ,2 hat i - hat j + hat k and 2 hat i + 5 hat j - hat k from a triangle in space.

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

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 vector 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 , 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 A and Bdot Find the position vectors of points of intersection.

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.

Find the angle between the lines vec r= dot( hat i+2 hat j- hat k)+lambda( hat i- hat j+ hat k) and the plane vec r . dot(3 hat i- hat j+ hat k)=4.