How can I know about matrices

Normalization of a vector

In order to be able to make measurements with vectors later, we have to use their amount To know.

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The amount of a vector or the length of the corresponding arrow is determined by $ | \ vec {v} | = \ sqrt {x_1 ^ 2 + x_2 ^ 2 + x_3 ^ 2} $.

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A vector $ \ vec {v} $ is called normalizedif it has the amount 1, i.e. if $ | \ vec {v} | = 1 $.

Any vector can be normalized by multiplying it by the reciprocal of its magnitude. Figuratively speaking, you divide by the "length" of your arrow. We mark a normalized vector with a small 0 as an index and thus write $ \ vec {v_0} $.

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The following applies: $ \ vec {v_0} = \ frac {1} {| \ vec {v} |} \ cdot \ vec {v} = \ frac {1} {\ sqrt {x_1 ^ 2 + x_2 ^ 2 + x_3 ^ 2}} \ cdot \ vec {v} $.

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The vector $ \ vec {v} = \ begin {pmatrix} 6 \ 3 \ 6 \ end {pmatrix} $ has the amount $ | \ vec {v} | = \ sqrt {36 + 9 + 36} = \ sqrt {81} = $ 9.
For the normalized vector $ \ vec {v_0} $, $ \ vec {v_0} = \ frac {1} {9} \ cdot \ vec {v} = \ frac {1} {9} \ cdot \ begin {pmatrix } 6 \ 3 \ 6 \ end {pmatrix} = \ begin {pmatrix} \ frac {2} {3} \ \ frac {1} {3} \ \ frac {2} {3} \ end { pmatrix} $.

In the video shown, the normalization of vectors is shown and explained again:

Video: normalization of a vector