The geometric definition is based on the notions of angle and distance (magnitude) of vectors.
Definisi geometris didasarkan pada pengertian sudut dan jarak (besaran vektor).
The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space.
Persamaan dua definisi ini bergantung pada memiliki sistem koordinat Kartesius untuk ruang Euklides.
In modern presentations of Euclidean geometry, the points of space are defined in terms of their Cartesian coordinates, and Euclidean space itself is commonly identified with the real coordinate space R n ^{n}} .
Dalam presentasi modern geometri Euclidean, titik-titik ruang ditentukan berdasarkan koordinat Cartesiannya,
[ 4 , − 2 , − 1 ]( 1 × 4 ) + ( 3 × − 2 ) + ( − 5 × − 1 )4 − 6 + 53 \ [1,3,-5]\cdot [4,-2,-1]&=(1\times 4)+(3\times -2)+(-5\times -1)\\&=4-6+5\\&=3\end{aligned}}}
[ 1 , 3 , − 5 ] ⋅ [ 4 , − 2 , − 1 ]( 1 ) ( 4 ) + ( 3 ) ( − 2 ) + ( − 5 ) ( − 1 )4 − 6 + 53. \ [1,3,-5]\cdot [4,-2,-1]&=(1)(4)+(3)(-2)+(-5)(-1)\\&=4-6+5\\&=3.\end{aligned}}} Dalam ruang Euclidean, suatu vektor Euclidean adalah sebuah objek geometri yang memiliki baik besaran (magnitude) dan arah (direction).
[ 1 , 3 , − 5 ]( 1 × 1 ) + ( 3 × 3 ) + ( − 5 × − 5 )1 + 9 + 2535 \ [1,3,-5]\cdot [1,3,-5]&=(1\times 1)+(3\times 3)+(-5\times -5)\\&=1+9+25\\&=35\end{aligned}}}
[ 1 , 3 , − 5 ] ⋅ [ 4 , − 2 , − 1 ]( 1 ) ( 4 ) + ( 3 ) ( − 2 ) + ( − 5 ) ( − 1 )4 − 6 + 53. \ [1,3,-5]\cdot [4,-2,-1]&=(1)(4)+(3)(-2)+(-5)(-1)\\&=4-6+5\\&=3.\end{aligned}}} Dalam ruang Euclidean, suatu vektor Euclidean adalah sebuah objek geometri yang memiliki baik besaran (magnitude) dan arah (direction).
The magnitude of a vector a } is denoted by ‖ a ‖ \right\|} .
Besaran vektor A dilambangkan dengan ‖ A ‖ \|} .
\cdot \mathbf {b}0.}
\cdot (\mathbf {b} +\mathbf {c} )
This implies that the dot product of a vector a } with itself is a ⋅ a‖ a ‖ 2 , \cdot \mathbf {a}\left\|\mathbf {a} \right\|^{2},} which gives ‖ a ‖a ⋅ a , \right\|={\sqrt {\mathbf {a} \cdot \mathbf {a} }},} the formula for the Euclidean length of the vector.
Ini menyiratkan bahwa produk skalar suatu vektor A dengan dirinya sendiri adalah A ⋅ A‖ A ‖ 2 , \cdot \mathbf {A}\|\mathbf {A} \|^{2},} yang menghasilkan ‖ A ‖A ⋅ A , \|={\sqrt {\mathbf {A} \cdot \mathbf {A} }},} rumus untuk panjang Euclidean vektor itu.
)\cdot \mathbf {b}\alpha (\mathbf {a} \cdot \mathbf {b} )
\cdot (\mathbf {b} +\mathbf {c} )
Commutative a ⋅ bb ⋅ a , \cdot \mathbf {b}\mathbf {b} \cdot \mathbf {a} ,} which follows from the definition ( θ is the angle between a } and b } ): a ⋅ b‖ a ‖ ‖ b ‖ cos ⁡ θ‖ b ‖ ‖ a ‖ cos ⁡ θb ⋅ a .
\cdot \mathbf {b}\mathbf {b} \cdot \mathbf {a} .} which follows from the definition (θ is the angle between a and b): a ⋅ b‖ a ‖ ‖ b ‖ cos ⁡ θ‖ b ‖ ‖ a ‖ cos ⁡ θb ⋅ a \cdot \mathbf {b}\|\mathbf {a} \|\|\mathbf {b} \|\cos \theta\|\mathbf {b} \|\|\mathbf {a} \|\cos \theta\mathbf {b} \cdot \mathbf {a} } Distributif over vector addition: a ⋅ ( b + c )
( c a ) ⋅ ba ⋅ ( c b ) .
a ⋅ b + a ⋅ c .
Orthogonal Two non-zero vectors a } and b } are orthogonal if and only if a ⋅ b0 \cdot \mathbf {b}0} .
=c_{1}c_{2}(\mathbf {a} \cdot \mathbf {b} )} Ortogonal: Dua vektor bukan-nol a dan b adalah ortogonal jika dan hanya jika a ⋅ b0.
If a ⋅ ba ⋅ c \cdot \mathbf {b}\mathbf {a} \cdot \mathbf {c} } and a ≠ 0 \neq \mathbf {0} } , then we can write: a ⋅ ( b − c )
° dan A ⋅ B‖ A ‖ ‖ B ‖ \cdot \mathbf {B}\|\mathbf {A} \|\,\|\mathbf {B} \|}
0 \cdot (\mathbf {b} -\mathbf {c} )
\cdot (\mathbf {b} +\mathbf {c} )
b ⋅ ( c × a )c ⋅ ( a × b ) .
a ⋅ b + a ⋅ c .
\cdot (\mathbf {b} \times \mathbf {c} )
\cdot (\mathbf {b} +\mathbf {c} )
=\mathbf {b} \cdot (\mathbf {c} \times \mathbf {a} )
=\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {c} .}
( a ⋅ c ) b − ( a ⋅ b ) c .
a ⋅ b + a ⋅ c .
\cdot \mathbf {b}\mathbf {b} ^{\mathsf {H}}\mathbf {a} .}
\cdot (\mathbf {b} +\mathbf {c} )