Show that the vectors `2hat(i)-hat(j)+hat(k), hat(i)-3hat(k)-5hat(k) " and " -2hat(i)+3hat(j)-4hat(k)` are the sides of a right-angled triangle.

Show that the vectors -hat(i)+hat(j), -4hat(i)-6hat(j) " and " 5hat(i)+5hat(j) are the sides of 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)+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.

Show that the poins whose position vectors are 4hat(i)+5hat(k),hat(i)+hat(j)+3hat(k)" and "-5hat(i)+3hat(j)-hat(k) are collinear.

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.

Show that the vectors a = 2 hat (i) + 3hat (j) + 6 hat (k) , b = 3 hat (i) - 6 hat (j) + 2 hat (k) and c= 6 hat (i) + 2 hat (j) - 3 hat (k) are mutually perpendicular

Show that the lines vec(r)=(hat(i)+hat(j)+hat(k))+t(hat(i)-hat(j)+hat(k)) and vec(r)=(3hat(i)-hat(k))+s(4hat(j)-16hat(k)) intersect and find the position vector of their point of intersection.

(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) .

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 vectors x hat (i) - 4 hat (j) + 5 hat (k) , hat (i) + 2 hat (j) + hat (k) and2 hat (i) - hat (j) + hat (k) are coplanar , find the value x