Show that is perpendicular to , for any two nonzero vectors

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Solution

The given vectors are | a → | b → +| b → | a → and | a → | b → −| b → | a → .

If two vectors are perpendicular to each other, then their dot product is zero.

The dot product of | a → | b → +| b → | a → and | a → | b → −| b → | a → is,

( | a → | b → +| b → | a → )⋅( | a → | b → −| b → | a → )=( | a → | b → )⋅( | a → | b → −| b → | a → )+( | b → | a → )⋅( | a → | b → −| b → | a → ) =( | a → | b → )⋅( | a → | b → )−( | a → | b → )⋅( | b → | a → )+( | b → | a → )⋅( | a → | b → )−( | b → | a → )⋅( | b → | a → ) = | a → | 2 | b → | 2 − | b → | 2 | a → | 2 =0

Hence, | a → | b → +| b → | a → is perpendicular to | a → | b → −| b → | a → .

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Show that is perpendicular to, for any two nonzero vectors

Show that $| a | b + | b | a$ is perpendicular to $| a | b - | b | a$ for any two non-zero vectors a and b.

a, b

a

b

a

r

r \cdot a = α,

α, a \times r = b

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