If four points A, B, C and D with position vectors and respectively are coplanar, then find the value of x.

For any two vectors and , show that:

If are three mutually perpendicular vectors of equal magnitude, prove that is equally inclined with vectors and . Also, find the angle.

Using vectors, prove that cos(A – B) = cosA cosB + sinA sinB.

For any two vectors and , prove that and hence show that (a₁b₁ + a₂b₂ + a₃b₃)² ≤ (a₁² + a₂² + a₃²)(b₁² + b₂² + b₃²).