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For the first question I really need all help I can get. Thanks!

1. Given a and b are unit vectors,

a) if the angle between them is 60 degrees, calculate (6a+b) . (a-2b)

b) if |a+b| = sqrt3, determine (2a-5b) . (b+3a)

2. The vectors a = 3i - 4j - k and b = 2i + 3j - 6k are the diagonals of a parallelogram. Show that this parallelogram is a rhombus, and determine the lengths of the sides and angles between the sides.

3. If a and b are perpendicular, show that |a|^2 + |b|^2 = |a + b|^2. What is the usual name of this result.

b) If a and b are not perpendicular, and a-b = c, express |c|^2 in terms of a and b. What is the usual name of this result?

You seem to post quite a few vector questions under the name of anonymous.
Are you the same person?
Please use a first name or some other nick to identify yourself.

Your first question uses the basic laws of vectors.

(2a-5b)∙(b+3a) = 2a∙b + 6│a│^2 - 5│b│^2 - 15a∙b

= 6│a│^2 - 5│b│^2 - 13a∙b

we know │a│ and │b│ are 1 each and
a∙b = │a││b│cos60º
a∙b = 1*1*1/2

so 6│a│^2 - 5│b│^2 - 13a∙b
= 6*1 - 5*1 - 13*1/2
= -11/2

for 1. b) make a diagram and find cosß using sides 1,1,√3
that way you can find a∙b and follow my example of a)

2. In a parallelogram the diagonals bisect each other, but in a rhombus (which is a parallelogram) they bisect each other at right angles.

so take the dot product, and see if you get zero. (you will)
then 1/2 of vector a + 1/2 of vector b will give you a side
etc.