Produk dot: Perbedaan antara revisi
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JohnThorne (bicara | kontrib) Tidak ada ringkasan suntingan |
JohnThorne (bicara | kontrib) Tidak ada ringkasan suntingan |
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Baris 82:
:<math>\mathbf A\cdot\mathbf B = \mathbf A\cdot\sum_i B_i\mathbf e_i = \sum_i B_i(\mathbf A\cdot\mathbf e_i) = \sum_i B_iA_i</math>
which is precisely the algebraic definition of the dot product. So the (geometric) dot product equals the (algebraic) dot product.
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== Sifat ==
▲The dot product fulfills the following properties if '''a''', '''b''', and '''c''' are real [[vector (geometry)|vectors]] and ''r'' is a [[scalar (mathematics)|scalar]].<ref name="Lipschutz2009" /><ref name="Spiegel2009" />
▲# '''[[Commutative]]:'''
#: <math> \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}.</math>
#: which follows from the definition (''θ'' is the angle between '''a''' and '''b'''):
#: <math>\mathbf{a}\cdot \mathbf{b} = \|\mathbf{a}\|\|\mathbf{b}\|\cos\theta = \|\mathbf{b}\|\|\mathbf{a}\|\cos\theta = \mathbf{b}\cdot\mathbf{a} </math>
# '''[[
#: <math> \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}. </math>
# '''[[bilinear form|Bilinear]]''':
Baris 96:
= r(\mathbf{a} \cdot \mathbf{b}) + (\mathbf{a} \cdot \mathbf{c}).
</math>
# '''[[
#: <math> (c_1\mathbf{a}) \cdot (c_2\mathbf{b}) = c_1 c_2 (\mathbf{a} \cdot \mathbf{b}) </math>
# '''[[
#:
# '''
#:
#:
# '''[[Product Rule]]:'''
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===Application to the cosine law===
[[File:Dot product cosine rule.svg|100px|thumb|Triangle with vector edges '''a''' and '''b''', separated by angle ''θ''.]]
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