Prove that 

(i) vector |a + b|² + vector |a + b|² = 2 vector { |a |² + |b |²}

 (ii) vector |a + b|² – vector |a + b|² = 4 vector (a • b)

If the position vectors of the points P and Q are 2 i + 3j + 4k and 3 i – 2 j – 3 k , find the direction cosines of the vector vector PQ and hence prove that,

Cos²α + Cos²β + Cos²γ = 1 

The vector −A is: 

A. greater than Vector A in magnitude 

B. less than Vector A in magnitude 

C. in the same direction as Vector A 

D. in the direction opposite to Vector A

E. perpendicular to vector A

If vector (a x b ) x c) = vector (a x ( b x c) , then prove that either vector a is parallel to vector c or vector b is perpendicular to both vector(a , c) 

Define a null vector ( or Zero vector ) 

If the magnitude of the sum of two vectors is less than the magnitude of either vector, then: 

A. the scalar product of the vectors must be negative 

B. the scalar product of the vectors must be positive 

C. the vectors must be parallel and in opposite directions

D. the vectors must be parallel and in the same direction 

E. none of the above

Prove that the vectors 3i –j – 2k & 2i -2j + 4k are orthogonal vectors. 

If the magnitude of the sum of two vectors is greater than the magnitude of either vector, then: 

A. the scalar product of the vectors must be negative 

B. the scalar product of the vectors must be positive 

C. the vectors must be parallel and in opposite directions 

D. the vectors must be parallel and in the same direction 

E. none of the above