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.

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.

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.

Given that the points with position vectors 12hat(i)-5hat(j), 10hat(i)+3hat(j) " and " xhat(i)+11hat(j) are collinear, find x.

(i) Determine the value of p for which the vectors phat(i)-5hat(j) " and " 2hat(i)-3hat(j) are collinear. (ii) The sides AB and BC of the triangle ABC are represented by the vectors 2hat(i)-hat(j)+2hat(k) " and " hat(i)+3hat(j)+5hat(k) respectively, find the vector representing side CA of the triangle ABC.

If the vector vec(AB) = 3hat(i) + 4hat(k) and vec(AC) = 5hat(i) - 2hat(j) + 4hat(k) are the sides of a triangle ABC, then the length of the median through A is-

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 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 .

Find a unit vector perpendicular to both the vectors 2 hat (i) - 3 hat (j) + 6 hat (k) and 3 hat (j) - 4 hat (k) . Also find the sine of the angle between the given vectors .

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 points having position vectors hat(i)+bhat(j)+chat(k), 7hat(i)+2hat(j)+6hat(k) " and " 5hat(i)+2hat(j)+5hat(k) are collinear, find the values of b and c.