The position vectors of the points A, B and C are `-5hat(i)+hat(j), 5hat(i)+5hat(j) " and " 10hat(i)+7hat(j)`, show that the points A, B, C are colinear.

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.

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.

(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) .

If the position vectors of the points A, B, C are -2hat(i)+2hat(j)+2hat(k), 2hat(i)+3hat(j)+3hat(k) " and " -hat(i)-2hat(j)+3hat(k) respectively, show that ABC is an isosceles triangle.

The position vectors of the points A and B are 3hat(i)-hat(j)+7hat(k) " and " 4hat(i)-3hat(j)-hat(k) , find the magnitude and direction cosines of vec(AB)

The position vectors of the points A, B, C and D are hat(i)+hat(j)+hat(k), 2hat(i)+3hat(j), 3hat(i)+5hat(j)-2hat(k) " and " -hat(j)+hat(k) respectively. Show that vec(AB) " and " vec(CD) are parallel and find the ratio of their moduli.

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 -

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-

If the position vectors of two given points A and B be 8hat(i)+3hat(j)+2hat(k) " and " 2hat(i)-5hat(j)+3hat(k) respectively, find the magnitude and direction of vec(AB) .