Given that the vectors
a,b, and
c each have a magnitude of
√2 and the angle between any two vectors is
‌, we can analyze the vectors
x and
y defined as follows:
|a|=|b|=|c|=√2First, let's express
x :
x=a×(b×c)Using the vector triple product identity:
a×(b×c)=(a⋅c)b−(a⋅b)cGiven that the angle between each pair of vectors is
‌, the dot products are:
a⋅b=|a||b|cos(‌)=√2×√2×‌=1Therefore, the expression for
x becomes:
x=1⋅b−1⋅c=b−cNow, consider
y :
y=b×(c×a)Using the vector triple product identity again:
b×(c×a)=(b⋅a)c−(b⋅c)aThis simplifies similarly to:
y=1⋅c−1⋅a=c−aHence, 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|