Vector Algebra

Given that the vectors a,b, and c each have a magnitude of √2 and the angle between any two vectors is ‌

π

3

, we can analyze the vectors x and y defined as follows:
|a|=|b|=|c|=√2
First, let's express x :
x=a×(b×c)
Using the vector triple product identity:
a×(b×c)=(a⋅c)b−(a⋅b)c
Given that the angle between each pair of vectors is ‌

π

3

, the dot products are:
a⋅b=|a||b|cos(‌

π

3

)=√2×√2×‌

1

2

=1
Therefore, the expression for x becomes:
x=1⋅b−1⋅c=b−c
Now, consider y :
y=b×(c×a)
Using the vector triple product identity again:
b×(c×a)=(b⋅a)c−(b⋅c)a
This simplifies similarly to:

y=1⋅c−1⋅a=c−a
Hence, the magnitudes of x and y are:
|x|‌=|b−c|
|y|‌=|c−a|
Since both expressions have the same form, it follows that:
|x|=|y|