Question

if a & b are perpendicular vectors, show that
a)(a+2b).(a-b)=|a|^2-2|b|^2
b)(a+b).(a-b)=|a|^2+|b|^2

*how to show it?just multiply it?if yes where does||come from?
does||show that it should be a value?

Answer 1

| | around a number means "magnitude of". You will need to use the following relationships:
a*a = |a|^2
b*b = |b|^2
a*b = 0, because they have said that a and b are perpendicular.

Th "*" operator means "dot product", which I assume you have learned the meaning of. In general, for vectors x and y,
x*y = |x| |y| cos(theta)
where theta is the angle between the two vectors. Thus, in this case,

a) (a+2b)*(a-b)=
a*a -a*b + 2b*a -2b*b

|a|^2 - 2|b|^2

Do problem (b) the same way.

Answer 2

My first line should have read:

| | around a number VECTOR means magnitude of that vector