If the position vector of A,B,C are 2i+4j-k,4i+5j+k and 3i+6j-3k respectively, then show that `triangleABC` is right angle.

If 4 hat i+7 hat j+8 hat k ,2 hat i+3 hat j+4hat ja n d2 hat i+5 hat j+7 hat k are the position vectors of the vertices A ,Ba n dC , respectively, of triangle A B C , then the position vecrtor of the point where the bisector of angle A meets B C is a. 2/3(-6 hat i-8 hat j- hat k) b. 2/3(6 hat i+8 hat j+6 hat k) c. 1/3(6 hat i+13 hat j+18 hat k) d. 1/3(5 hat j+12 hat k)

The position vectors of two masses of 6 and 2 units are 6i-7j and 2i+5j-8k respectively. Deduce the position of their centre of mass.

If the position vectors of the points P and Q are 2hat(i)+hat(k) " and " -3hat(i)-4hat(j)-5hat(k) respectively, then vector vec(OP) is -

Find a unit vector perpendicular to the plane of ABC where the position vectors of A,B,C are 3i-j+2k,i-j-3k and 4i-3j+k respectively.

In a DeltaABC , right angled at the vertex A, If the positive vectors of A, B and C are respectively 3hat(i) + hat(j) - hat(k), -hat(i) + 3hat(j) + phat(k) and 5hat(i) + qhat(j) - 4hat(k) , then the point (p,q) lies on a line-

Show that the point A , B and C with position vectors vec a =3 hat i - 4 hat j -4 hat k vec b = 2 hat i - hat j + hat k and vec c = hat i - 3 hat j - 5 hat k , respectively form the vertices of a right angled triangle.

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.

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.

(i) If the position vectors of the points A, B, C be 5hat(i)+3hat(j)+4hat(k), hat(i)+5hat(j)+hat(k) " and " -3hat(i)+7hat(j)-2hat(k) respectively, then show that the points B bisects the line-segment bar(AC) . (ii) The position vectors of the points P and Q are 5hat(i)-12hat(j)+5hat(k) " and " -4hat(i)+3hat(j)-hat(k) respectively. Find the position vectors of the trisection points of the line-segment bar(PQ) .

The position vectors of four points A,B,C and D are given below . In each case , using vector method prove that the four points A,B,C and D are coplanar . 4 hat (i) + 8 hat (j) + 12 hat (k) , 2 hat (i) + 4 hat (j) + 6 hat(k), 3 hat (i) + 5 hat (j) + 4 hat(k) , 5 hat (i) + 8hat (j) + 5 hat (k)