Show that the points A(2i^−j^+k^),B(i^−3j^−5k^),C(3i^−4j^−4k^)are the vertices of a right angled triangle.
OABCDE is a regular hexagon of side 2 units in the XY-plane in the first quadrant. O being the origin and OA taken along the x-axis. A point P is taken on a line parallel to the z-axis through the centre of the hexagon at a distance of 3 unit from O in the positive Z direction. Then find vector AP.
Statement 1:Let A(a),B(b)andC(c) be three points such that a=2i^+k^,b=3i^−j^+3k^andc=−i^+7j^−5k^˙ Then OABC is a tetrahedron. Statement 2: Let A(a),B(b)andC(c) be three points such that vectors a,bandc are non-coplanar. Then OABC is a tetrahedron where O is the origin.
If AndB are two vectors and k any scalar quantity greater than zero, then prove that ∣∣A+B∣∣2≤(1+k)∣∣A∣∣2+(1+k1)∣∣B∣∣2˙
Show that the points A(1,−2,−8),B(5,0,−2)andC(1,3,7) are collinear, and find the ratio in which B divides AC˙