ডট গুণন নিম্নলিখিত বৈশিষ্ট্য প্রদর্শন করে, যদি
a
{\displaystyle \mathbf {a} }
b
{\displaystyle \mathbf {b} }
c
{\displaystyle \mathbf {c} }
ভেক্টর এবং
r
{\displaystyle r}
c
1
{\displaystyle c_{1}}
c
2
{\displaystyle c_{2}}
স্কেলার হয় [১] [২]
বিনিময় সূত্র
a
⋅
b
=
b
⋅
a
,
{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {b} \cdot \mathbf {a} ,}
a
{\displaystyle \mathbf {a} }
b
{\displaystyle \mathbf {b} }
θ
{\displaystyle \theta }
[৩]
a
⋅
b
=
‖
a
‖
‖
b
‖
cos
θ
=
‖
b
‖
‖
a
‖
cos
θ
=
b
⋅
a
.
{\displaystyle \mathbf {a} \cdot \mathbf {b} =\left\|\mathbf {a} \right\|\left\|\mathbf {b} \right\|\cos \theta =\left\|\mathbf {b} \right\|\left\|\mathbf {a} \right\|\cos \theta =\mathbf {b} \cdot \mathbf {a} .}
a
⋅
(
b
+
c
)
=
a
⋅
b
+
a
⋅
c
.
{\displaystyle \mathbf {a} \cdot (\mathbf {b} +\mathbf {c} )=\mathbf {a} \cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {c} .}
a
⋅
(
r
b
+
c
)
=
r
(
a
⋅
b
)
+
(
a
⋅
c
)
.
{\displaystyle \mathbf {a} \cdot (r\mathbf {b} +\mathbf {c} )=r(\mathbf {a} \cdot \mathbf {b} )+(\mathbf {a} \cdot \mathbf {c} ).}
স্কেলার গুন
(
c
1
a
)
⋅
(
c
2
b
)
=
c
1
c
2
(
a
⋅
b
)
.
{\displaystyle (c_{1}\mathbf {a} )\cdot (c_{2}\mathbf {b} )=c_{1}c_{2}(\mathbf {a} \cdot \mathbf {b} ).}
ইহা গ্রহনযোগ্য নয়
কারণ একটি স্কেলার
a
⋅
b
{\displaystyle \mathbf {a} \cdot \mathbf {b} }
c
{\displaystyle \mathbf {c} }
(
a
⋅
b
)
⋅
c
{\displaystyle (\mathbf {a} \cdot \mathbf {b} )\cdot \mathbf {c} }
a
⋅
(
b
⋅
c
)
{\displaystyle \mathbf {a} \cdot (\mathbf {b} \cdot \mathbf {c} )}
[৪] [৫]
c
(
a
⋅
b
)
=
(
c
a
)
⋅
b
=
a
⋅
(
c
b
)
{\displaystyle c(\mathbf {a} \cdot \mathbf {b} )=(c\mathbf {a} )\cdot \mathbf {b} =\mathbf {a} \cdot (c\mathbf {b} )}
[৬]
লম্ব দুটি অশূন্য ভেক্টর
a
{\displaystyle \mathbf {a} }
b
{\displaystyle \mathbf {b} }
a
⋅
b
=
0
{\displaystyle \mathbf {a} \cdot \mathbf {b} =0}
সাধারণ সংখ্যার গুণের বিপরীতে, যদি
a
b
=
a
c
{\displaystyle ab=ac}
b
{\displaystyle b}
c
{\displaystyle c}
a
{\displaystyle a}
যদি
a
⋅
b
=
a
⋅
c
{\displaystyle \mathbf {a} \cdot \mathbf {b} =\mathbf {a} \cdot \mathbf {c} }
a
≠
0
{\displaystyle \mathbf {a} \neq \mathbf {0} }
a
⋅
(
b
−
c
)
=
0
{\displaystyle \mathbf {a} \cdot (\mathbf {b} -\mathbf {c} )=0}
a
{\displaystyle \mathbf {a} }
(
b
−
c
)
{\displaystyle (\mathbf {b} -\mathbf {c} )}
(
b
−
c
)
≠
0
{\displaystyle (\mathbf {b} -\mathbf {c} )\neq \mathbf {0} }
b
≠
c
{\displaystyle \mathbf {b} \neq \mathbf {c} }
গুণের সূত্র যদি
a
{\displaystyle \mathbf {a} }
b
{\displaystyle \mathbf {b} }
′
{\displaystyle {}'}
a
⋅
b
{\displaystyle \mathbf {a} \cdot \mathbf {b} }
(
a
⋅
b
)
′
=
a
′
⋅
b
+
a
⋅
b
′
.
{\displaystyle (\mathbf {a} \cdot \mathbf {b} )'=\mathbf {a} '\cdot \mathbf {b} +\mathbf {a} \cdot \mathbf {b} '.}
ত্রিভুজের a এবং b ভেক্টর , কোণ θ দ্বারা বিভক্ত। উপরোক্ত ভেক্টর দ্বয়
a
{\displaystyle {\color {red}\mathbf {a} }}
b
{\displaystyle {\color {blue}\mathbf {b} }}
θ
{\displaystyle \theta }
c
=
a
−
b
{\displaystyle {\color {orange}\mathbf {c} }={\color {red}\mathbf {a} }-{\color {blue}\mathbf {b} }}
a
{\displaystyle a}
b
{\displaystyle b}
c
{\displaystyle c}
a
{\displaystyle {\color {red}\mathbf {a} }}
b
{\displaystyle {\color {blue}\mathbf {b} }}
c
{\displaystyle {\color {orange}\mathbf {c} }}
এটি কোসাইনের নিয়ম ।
↑ S. Lipschutz; M. Lipson (২০০৯)। Linear Algebra (Schaum's Outlines) (4th সংস্করণ)। McGraw Hill। আইএসবিএন 978-0-07-154352-1 । ↑ M.R. Spiegel; S. Lipschutz (২০০৯)। Vector Analysis (Schaum's Outlines) (2nd সংস্করণ)। McGraw Hill। আইএসবিএন 978-0-07-161545-7 । ↑ Nykamp, Duane। "The dot product" । Math Insight । সংগ্রহের তারিখ সেপ্টেম্বর ৬, ২০২০ । ↑ Weisstein, Eric W. "Dot Product." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DotProduct.html
↑ T. Banchoff; J. Wermer (১৯৮৩)। Linear Algebra Through Geometry আইএসবিএন 978-1-4684-0161-5 । ↑ A. Bedford; Wallace L. Fowler (২০০৮)। Engineering Mechanics: Statics (5th সংস্করণ)। Prentice Hall। পৃষ্ঠা 60। আইএসবিএন 978-0-13-612915-8 ।
