The vector is and the vector is .

The projection of a vector onto the vector is , where is a real number.

The projection of the vector onto the vector is , where is a real number.

Find the vector in terms of and .   (4 marks)

Given the two non-zero vectors and , let be the projection of onto .

What is the projection of onto ?

The vectors and are not parallel. The vector is the projection of onto the vector .

The vector is parallel to so it can be written for some real number . (Do NOT prove this.)

Prove that    is smallest when by showing that, for all real numbers .  (3 marks)

The following diagram shows the vector and the vectors and . 
 

Which statement regarding this diagram could be true?

1. The projection of onto    is the vector  .
2. The projection of onto    is the vector  .
3. The projection of onto    is the vector  . 
4. The projection of onto    is the vector  . 

Consider the vectors,    where    and    where  .

If  , find    as a multiple of  .  (2 marks)

Given    and  , what is the magnitude of the projection of    onto  . Give your answer in simplest form.  (3 marks)

Find the projection of    onto    given    and  .  (2 marks)