a
→
=
(
a
1
,
a
2
)
=
(
a
1
a
2
)
=
a
1
i
^
+
a
2
j
^
{\displaystyle {\vec {a}}=(a_{1},a_{2})={\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}=a_{1}{\hat {i}}+a_{2}{\hat {j}}}
a
→
=
(
a
1
,
a
2
,
a
3
)
=
(
a
1
a
2
a
3
)
=
a
1
i
^
+
a
2
j
^
+
a
3
k
^
{\displaystyle {\vec {a}}=(a_{1},a_{2},a_{3})={\begin{pmatrix}a_{1}\\a_{2}\\a_{3}\end{pmatrix}}=a_{1}{\hat {i}}+a_{2}{\hat {j}}+a_{3}{\hat {k}}}
Panjang vektor
sunting
Berada di
R
2
{\displaystyle R^{2}}
Panjang vektor a dalam posisi
(
a
1
,
a
2
)
{\displaystyle (a_{1},a_{2})}
adalah
|
a
→
|
=
a
1
2
+
a
2
2
{\displaystyle \left|{\vec {a}}\right|={\sqrt {a_{1}^{2}+a_{2}^{2}}}}
Panjang vektor b dalam posisi
(
b
1
,
b
2
)
{\displaystyle (b_{1},b_{2})}
adalah
|
b
→
|
=
b
1
2
+
b
2
2
{\displaystyle \left|{\vec {b}}\right|={\sqrt {b_{1}^{2}+b_{2}^{2}}}}
Panjang vektor c dalam posisi
(
a
1
,
a
2
)
{\displaystyle (a_{1},a_{2})}
dan
(
b
1
,
b
2
)
{\displaystyle (b_{1},b_{2})}
adalah
|
c
→
|
=
(
b
1
−
a
1
)
2
+
(
b
2
−
a
2
)
2
{\displaystyle \left|{\vec {c}}\right|={\sqrt {(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}}}}
Berada di
R
3
{\displaystyle R^{3}}
Panjang vektor a dalam posisi
(
a
1
,
a
2
,
a
3
)
{\displaystyle (a_{1},a_{2},a_{3})}
adalah
|
a
→
|
=
a
1
2
+
a
2
2
+
a
3
2
{\displaystyle \left|{\vec {a}}\right|={\sqrt {a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}}}
Panjang vektor b dalam posisi
(
b
1
,
b
2
,
b
3
)
{\displaystyle (b_{1},b_{2},b_{3})}
adalah
|
b
→
|
=
b
1
2
+
b
2
2
+
b
3
2
{\displaystyle \left|{\vec {b}}\right|={\sqrt {b_{1}^{2}+b_{2}^{2}+b_{3}^{2}}}}
Panjang vektor c dalam posisi
(
a
1
,
a
2
,
a
3
)
{\displaystyle (a_{1},a_{2},a_{3})}
dan
(
b
1
,
b
2
,
b
3
)
{\displaystyle (b_{1},b_{2},b_{3})}
adalah
|
c
→
|
=
(
b
1
−
a
1
)
2
+
(
b
2
−
a
2
)
2
+
(
b
3
−
a
3
)
2
{\displaystyle \left|{\vec {c}}\right|={\sqrt {(b_{1}-a_{1})^{2}+(b_{2}-a_{2})^{2}+(b_{3}-a_{3})^{2}}}}
Jumlah dan selisih kedua vektor
|
a
→
±
b
→
|
=
|
a
→
|
2
+
|
b
→
|
2
±
2
a
→
⋅
b
→
⋅
c
o
s
C
{\displaystyle \left|{\vec {a}}\pm {\vec {b}}\right|={\sqrt {|{\vec {a}}|^{2}+|{\vec {b}}|^{2}\pm 2{\vec {a}}\cdot {\vec {b}}\cdot cosC}}}
a
^
=
a
→
|
a
→
|
{\displaystyle {\hat {a}}={\frac {\vec {a}}{\left|{\vec {a}}\right|}}}
Operasi aljabar pada vektor
sunting
Penjumlahan dan pengurangan
terdiri dari 2 aturan jenis yaitu aturan segitiga dan jajar genjang
c
→
=
a
→
+
b
→
=
(
a
1
a
2
)
+
(
b
1
b
2
)
=
(
a
1
+
b
1
a
2
+
b
2
)
{\displaystyle {\vec {c}}={\vec {a}}+{\vec {b}}={\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}+{\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}={\begin{pmatrix}{a_{1}+b_{1}}\\{a_{2}+b_{2}}\end{pmatrix}}}
c
→
=
a
→
−
b
→
=
(
a
1
a
2
)
−
(
b
1
b
2
)
=
(
a
1
−
b
1
a
2
−
b
2
)
{\displaystyle {\vec {c}}={\vec {a}}-{\vec {b}}={\begin{pmatrix}a_{1}\\a_{2}\end{pmatrix}}-{\begin{pmatrix}b_{1}\\b_{2}\end{pmatrix}}={\begin{pmatrix}{a_{1}-b_{1}}\\{a_{2}-b_{2}}\end{pmatrix}}}
skalar dengan vektor
Jika k skalar tak nol dan vektor
a
→
=
a
1
i
^
+
a
2
j
^
+
a
3
k
^
{\displaystyle {\vec {a}}=a_{1}{\hat {i}}+a_{2}{\hat {j}}+a_{3}{\hat {k}}}
maka vektor
k
a
→
=
(
k
a
1
,
k
a
2
,
k
a
3
)
{\displaystyle k{\vec {a}}=(ka_{1},ka_{2},ka_{3})}
titik dua vektor
Jika vektor
a
→
=
a
1
i
^
+
a
2
j
^
+
a
3
k
^
{\displaystyle {\vec {a}}=a_{1}{\hat {i}}+a_{2}{\hat {j}}+a_{3}{\hat {k}}}
dan vektor
b
→
=
b
1
i
^
+
b
2
j
^
+
b
3
k
^
{\displaystyle {\vec {b}}=b_{1}{\hat {i}}+b_{2}{\hat {j}}+b_{3}{\hat {k}}}
maka
a
→
⋅
b
→
=
a
1
b
1
+
a
2
b
2
+
a
3
b
3
{\displaystyle {\vec {a}}\cdot {\vec {b}}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}}
titik dua vektor dengan membentuk sudut
Jika
a
→
{\displaystyle {\vec {a}}}
dan
b
→
{\displaystyle {\vec {b}}}
vektor tak nol dan sudut
α
{\displaystyle \alpha }
diantara vektor
a
→
{\displaystyle {\vec {a}}}
dan
b
→
{\displaystyle {\vec {b}}}
maka perkalian skalar vektor
a
→
{\displaystyle {\vec {a}}}
dan
b
→
{\displaystyle {\vec {b}}}
adalah
a
→
⋅
b
→
{\displaystyle {\vec {a}}\cdot {\vec {b}}}
=
|
a
→
|
⋅
|
b
→
|
c
o
s
α
{\displaystyle \left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|cos\alpha }
silang dua vektor
Jika vektor
a
→
=
a
1
i
^
+
a
2
j
^
+
a
3
k
^
{\displaystyle {\vec {a}}=a_{1}{\hat {i}}+a_{2}{\hat {j}}+a_{3}{\hat {k}}}
dan vektor
b
→
=
b
1
i
^
+
b
2
j
^
+
b
3
k
^
{\displaystyle {\vec {b}}=b_{1}{\hat {i}}+b_{2}{\hat {j}}+b_{3}{\hat {k}}}
maka
a
→
×
b
→
=
(
a
2
b
3
i
^
+
a
3
b
1
j
^
+
a
1
b
2
k
^
)
−
(
a
2
b
1
k
^
+
a
3
b
2
i
^
+
a
1
b
3
j
^
)
{\displaystyle {\vec {a}}\times {\vec {b}}=(a_{2}b_{3}{\hat {i}}+a_{3}b_{1}{\hat {j}}+a_{1}b_{2}{\hat {k}})-(a_{2}b_{1}{\hat {k}}+a_{3}b_{2}{\hat {i}}+a_{1}b_{3}{\hat {j}})}
[
i
^
j
^
k
^
i
^
j
^
a
1
a
2
a
3
a
1
a
2
b
1
b
2
b
3
b
1
b
2
−
−
−
+
+
+
]
{\displaystyle \left[{\begin{array}{rrr|rr}{\hat {i}}&{\hat {j}}&{\hat {k}}&{\hat {i}}&{\hat {j}}\\a_{1}&a_{2}&a_{3}&a_{1}&a_{2}\\b_{1}&b_{2}&b_{3}&b_{1}&b_{2}\\-&-&-+&+&+\\\end{array}}\right]}
silang dua vektor dengan membentuk sudut
Jika
a
→
{\displaystyle {\vec {a}}}
dan
b
→
{\displaystyle {\vec {b}}}
vektor tak nol dan sudut
α
{\displaystyle \alpha }
diantara vektor
a
→
{\displaystyle {\vec {a}}}
dan
b
→
{\displaystyle {\vec {b}}}
maka perkalian skalar vektor
a
→
{\displaystyle {\vec {a}}}
dan
b
→
{\displaystyle {\vec {b}}}
adalah
a
→
×
b
→
{\displaystyle {\vec {a}}\times {\vec {b}}}
=
|
a
→
|
×
|
b
→
|
s
i
n
α
{\displaystyle \left|{\vec {a}}\right|\times \left|{\vec {b}}\right|sin\alpha }
Sifat operasi aljabar pada vektor
sunting
a
→
+
b
→
=
b
→
+
a
→
{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}}
(
a
→
+
b
→
)
+
c
→
=
a
→
+
(
b
→
+
c
→
)
{\displaystyle ({\vec {a}}+{\vec {b}})+{\vec {c}}={\vec {a}}+({\vec {b}}+{\vec {c}})}
a
→
+
0
=
0
+
a
→
{\displaystyle {\vec {a}}+0=0+{\vec {a}}}
k
(
a
→
+
b
→
)
=
k
a
→
+
k
b
→
{\displaystyle k({\vec {a}}+{\vec {b}})=k{\vec {a}}+k{\vec {b}}}
(
k
+
l
)
a
→
=
k
a
→
+
l
a
→
{\displaystyle (k+l){\vec {a}}=k{\vec {a}}+l{\vec {a}}}
a
→
+
(
−
a
→
)
=
0
{\displaystyle {\vec {a}}+(-{\vec {a}})=0}
a
→
⋅
b
→
=
b
→
⋅
a
→
{\displaystyle {\vec {a}}\cdot {\vec {b}}={\vec {b}}\cdot {\vec {a}}}
(
a
→
⋅
b
→
)
⋅
c
→
=
a
→
⋅
(
b
→
⋅
c
→
)
{\displaystyle ({\vec {a}}\cdot {\vec {b}})\cdot {\vec {c}}={\vec {a}}\cdot ({\vec {b}}\cdot {\vec {c}})}
a
→
⋅
1
=
1
⋅
a
→
{\displaystyle {\vec {a}}\cdot 1=1\cdot {\vec {a}}}
k
(
a
→
⋅
b
→
)
=
k
a
→
⋅
b
→
=
a
→
⋅
k
b
→
{\displaystyle k({\vec {a}}\cdot {\vec {b}})=k{\vec {a}}\cdot {\vec {b}}={\vec {a}}\cdot k{\vec {b}}}
(
k
⋅
l
)
a
→
=
k
(
l
⋅
a
→
)
{\displaystyle (k\cdot l){\vec {a}}=k(l\cdot {\vec {a}})}
a
→
⋅
a
→
=
|
a
→
|
2
{\displaystyle {\vec {a}}\cdot {\vec {a}}=\left|{\vec {a}}\right|^{2}}
a
→
×
b
→
≠
b
→
×
a
→
{\displaystyle {\vec {a}}\times {\vec {b}}\neq {\vec {b}}\times {\vec {a}}}
a
→
×
b
→
=
−
(
b
→
×
a
→
)
{\displaystyle {\vec {a}}\times {\vec {b}}=-({\vec {b}}\times {\vec {a}})}
(
a
→
×
b
→
)
×
c
→
≠
a
→
×
(
b
→
×
c
→
)
{\displaystyle ({\vec {a}}\times {\vec {b}})\times {\vec {c}}\neq {\vec {a}}\times ({\vec {b}}\times {\vec {c}})}
a
→
⋅
(
b
→
×
c
→
)
=
b
→
⋅
(
c
→
×
a
→
)
=
c
→
⋅
(
a
→
×
b
→
)
{\displaystyle {\vec {a}}\cdot ({\vec {b}}\times {\vec {c}})={\vec {b}}\cdot ({\vec {c}}\times {\vec {a}})={\vec {c}}\cdot ({\vec {a}}\times {\vec {b}})}
a
→
×
(
b
→
+
c
→
)
=
a
→
×
b
→
+
a
→
×
c
→
{\displaystyle {\vec {a}}\times ({\vec {b}}+{\vec {c}})={\vec {a}}\times {\vec {b}}+{\vec {a}}\times {\vec {c}}}
k
(
a
→
×
b
→
)
=
k
a
→
×
b
→
=
a
→
×
k
b
→
{\displaystyle k({\vec {a}}\times {\vec {b}})=k{\vec {a}}\times {\vec {b}}={\vec {a}}\times k{\vec {b}}}
Hubungan vektor dengan vektor lain
sunting
Saling tegak lurus
Jika tegak lurus antara vektor
a
→
{\displaystyle {\vec {a}}}
dengan vektor
b
→
{\displaystyle {\vec {b}}}
maka
a
→
⋅
b
→
=
|
a
→
|
⋅
|
b
→
|
cos
90
∘
{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\cos {90}^{\circ }}
a
→
⋅
b
→
=
0
{\displaystyle {\vec {a}}\cdot {\vec {b}}=0}
Sejajar
Jika vektor
a
→
{\displaystyle {\vec {a}}}
sejajar dengan vektor
b
→
{\displaystyle {\vec {b}}}
maka
a
→
⋅
b
→
=
|
a
→
|
⋅
|
b
→
|
cos
0
∘
{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\cos {0}^{\circ }}
a
→
⋅
b
→
=
|
a
→
|
⋅
|
b
→
|
{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}
a
→
⋅
b
→
=
|
a
→
|
⋅
|
b
→
|
cos
180
∘
{\displaystyle {\vec {a}}\cdot {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\cos {180}^{\circ }}
a
→
⋅
b
→
=
−
|
a
→
|
⋅
|
b
→
|
{\displaystyle {\vec {a}}\cdot {\vec {b}}=-\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}
Saling tegak lurus
Jika tegak lurus antara vektor
a
→
{\displaystyle {\vec {a}}}
dengan vektor
b
→
{\displaystyle {\vec {b}}}
maka
a
→
×
b
→
=
|
a
→
|
⋅
|
b
→
|
sin
90
∘
{\displaystyle {\vec {a}}\times {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\sin {90}^{\circ }}
a
→
×
b
→
=
|
a
→
|
⋅
|
b
→
|
{\displaystyle {\vec {a}}\times {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}
Gagal mengurai (SVG (MathML dapat diaktifkan melalui plugin peramban): Respons tak sah ("Math extension cannot connect to Restbase.") dari peladen "http://localhost:6011/id.wikibooks.org/v1/":): {\displaystyle \vec a \times \vec b = \left| \vec a \right| \cdot \left| \vec b \right| \sin {270}^{\circ}}
a
→
×
b
→
=
−
|
a
→
|
⋅
|
b
→
|
{\displaystyle {\vec {a}}\times {\vec {b}}=-\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}
Jika
β
>
90
∘
{\displaystyle \beta >{90}^{\circ }}
maka dua vektor tersebut searah
Jika
β
<
90
∘
{\displaystyle \beta <{90}^{\circ }}
maka vektor saling berlawanan arah
Sejajar
Jika vektor
a
→
{\displaystyle {\vec {a}}}
sejajar dengan vektor
b
→
{\displaystyle {\vec {b}}}
maka
a
→
×
b
→
=
|
a
→
|
⋅
|
b
→
|
sin
0
∘
{\displaystyle {\vec {a}}\times {\vec {b}}=\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|\sin {0}^{\circ }}
a
→
×
b
→
=
0
{\displaystyle {\vec {a}}\times {\vec {b}}=0}
Sudut dua vektor
sunting
Jika vektor
a
→
{\displaystyle {\vec {a}}}
dan vektor
b
→
{\displaystyle {\vec {b}}}
sudut yang dapat dibentuk dari kedua vektor tersebut adalah
c
o
s
α
=
a
→
⋅
b
→
|
a
→
|
⋅
|
b
→
|
{\displaystyle cos\alpha ={\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {a}}\right|\cdot \left|{\vec {b}}\right|}}}
Panjang proyeksi dan proyeksi vektor
sunting
Panjang proyeksi vektor
a
→
{\displaystyle {\vec {a}}}
pada vektor
b
→
{\displaystyle {\vec {b}}}
adalah
|
c
→
|
=
a
→
⋅
b
→
|
b
→
|
{\displaystyle \left|{\vec {c}}\right|={\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {b}}\right|}}}
Proyeksi vektor
a
→
{\displaystyle {\vec {a}}}
pada vektor
b
→
{\displaystyle {\vec {b}}}
adalah
c
→
=
a
→
⋅
b
→
|
b
→
|
2
⋅
b
→
{\displaystyle {\vec {c}}={\frac {{\vec {a}}\cdot {\vec {b}}}{\left|{\vec {b}}\right|^{2}}}\cdot {\vec {b}}}
segitiga
R
→
=
a
→
+
b
→
{\displaystyle {\vec {R}}={\vec {a}}+{\vec {b}}}
jajar genjang
R
→
=
|
a
→
−
b
→
|
=
|
a
→
|
2
+
|
b
→
|
2
−
2
⋅
a
→
⋅
b
→
⋅
c
o
s
C
{\displaystyle {\vec {R}}=|{\vec {a}}-{\vec {b}}|={\sqrt {|{\vec {a}}|^{2}+|{\vec {b}}|^{2}-2\cdot {\vec {a}}\cdot {\vec {b}}\cdot cosC}}}
Aturan jajar genjang
Posisi vektor
N
→
=
m
s
+
n
r
m
+
n
{\displaystyle {\vec {N}}={\frac {ms+nr}{m+n}}}
Berada di
R
2
{\displaystyle R^{2}}
N
→
=
(
m
x
2
+
n
x
1
m
+
n
,
m
y
2
+
n
y
1
m
+
n
)
{\displaystyle {\vec {N}}=({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}})}
Berada di
R
3
{\displaystyle R^{3}}
N
→
=
(
m
x
2
+
n
x
1
m
+
n
,
m
y
2
+
n
y
1
m
+
n
,
m
z
2
+
n
z
1
m
+
n
)
{\displaystyle {\vec {N}}=({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}},{\frac {mz_{2}+nz_{1}}{m+n}})}
Satu garis
Perbandingan posisi dalam adalah m:n
Posisi vektor
N
→
=
m
s
+
n
r
m
+
n
{\displaystyle {\vec {N}}={\frac {ms+nr}{m+n}}}
Berada di
R
2
{\displaystyle R^{2}}
N
→
=
(
m
x
2
+
n
x
1
m
+
n
,
m
y
2
+
n
y
1
m
+
n
)
{\displaystyle {\vec {N}}=({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}})}
Berada di
R
3
{\displaystyle R^{3}}
N
→
=
(
m
x
2
+
n
x
1
m
+
n
,
m
y
2
+
n
y
1
m
+
n
,
m
z
2
+
n
z
1
m
+
n
)
{\displaystyle {\vec {N}}=({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}},{\frac {mz_{2}+nz_{1}}{m+n}})}
Perbandingan posisi luar adalah m:-n
Posisi vektor
N
→
=
m
s
−
n
r
m
−
n
{\displaystyle {\vec {N}}={\frac {ms-nr}{m-n}}}
Berada di
R
2
{\displaystyle R^{2}}
N
→
=
(
m
x
2
−
n
x
1
m
−
n
,
m
y
2
−
n
y
1
m
−
n
)
{\displaystyle {\vec {N}}=({\frac {mx_{2}-nx_{1}}{m-n}},{\frac {my_{2}-ny_{1}}{m-n}})}
Berada di
R
3
{\displaystyle R^{3}}
N
→
=
(
m
x
2
−
n
x
1
m
−
n
,
m
y
2
−
n
y
1
m
−
n
,
m
z
2
−
n
z
1
m
−
n
)
{\displaystyle {\vec {N}}=({\frac {mx_{2}-nx_{1}}{m-n}},{\frac {my_{2}-ny_{1}}{m-n}},{\frac {mz_{2}-nz_{1}}{m-n}})}
contoh
Titik a -3i-2j+4k dan b 6i+6j+k. tentukan:
a
→
+
b
→
{\displaystyle {\vec {a}}+{\vec {b}}}
a
→
−
b
→
{\displaystyle {\vec {a}}-{\vec {b}}}
a
→
⋅
b
→
{\displaystyle {\vec {a}}\cdot {\vec {b}}}
a
→
×
b
→
{\displaystyle {\vec {a}}\times {\vec {b}}}
panjang vektor
vektor satuan pada vektor b
panjang proyeksi
proyeksi vektor
Jawaban
∗
c
→
=
a
→
+
b
→
=
(
−
3
,
−
2
,
4
)
+
(
6
,
6
,
1
)
=
(
−
3
+
6
)
,
(
−
2
+
6
)
,
(
4
+
1
)
=
3
,
4
,
5
∗
c
→
=
a
→
−
b
→
=
(
−
3
,
−
2
,
4
)
−
(
6
,
6
,
1
)
=
(
−
3
−
6
)
,
(
−
2
−
6
)
,
(
4
−
1
)
=
−
9
,
−
8
,
3
∗
a
→
⋅
b
→
=
(
−
3
,
−
2
,
4
)
⋅
(
6
,
6
,
1
)
=
(
−
3
⋅
6
)
+
(
−
2
⋅
6
)
+
(
4
⋅
1
)
=
−
9
−
8
+
3
=
−
14
∗
a
→
×
b
→
i
j
k
i
j
−
3
−
2
4
−
3
−
2
6
6
1
6
6
−
−
−
+
+
+
=
(
−
2
⋅
1
)
i
+
(
4
⋅
6
)
j
+
(
−
3
⋅
6
)
k
−
(
−
3
⋅
1
)
j
−
(
4
⋅
6
)
i
−
(
−
2
⋅
6
)
k
=
−
2
i
+
24
j
−
18
k
+
3
j
−
24
i
+
12
k
=
−
26
i
+
27
j
−
6
k
∗
|
c
→
|
=
(
6
−
(
−
3
)
)
2
+
(
6
−
(
−
2
)
)
2
+
(
1
−
4
)
2
=
9
2
+
8
2
+
(
−
3
)
2
=
81
+
64
+
9
=
81
+
64
+
9
=
145
∗
b
^
=
b
→
|
b
→
|
=
6
,
6
,
1
6
2
+
6
2
+
1
2
=
6
,
6
,
1
36
+
36
+
1
=
6
,
6
,
1
73
∗
|
c
→
|
=
a
→
⋅
b
→
|
b
→
|
=
14
73
∗
c
→
=
a
→
⋅
b
→
|
b
→
|
2
⋅
b
→
=
14
(
73
)
2
(
6
,
6
,
1
)
=
14
73
(
6
,
6
,
1
)
{\displaystyle {\begin{aligned}*{\vec {c}}&={\vec {a}}+{\vec {b}}\\&=(-3,-2,4)+(6,6,1)\\&=(-3+6),(-2+6),(4+1)\\&=3,4,5\\*{\vec {c}}&={\vec {a}}-{\vec {b}}\\&=(-3,-2,4)-(6,6,1)\\&=(-3-6),(-2-6),(4-1)\\&=-9,-8,3\\*{\vec {a}}\cdot {\vec {b}}\\&=(-3,-2,4)\cdot (6,6,1)\\&=(-3\cdot 6)+(-2\cdot 6)+(4\cdot 1)\\&=-9-8+3\\&=-14\\*{\vec {a}}\times {\vec {b}}\\{\begin{array}{rrr|rr}i&j&k&i&j\\-3&-2&4&-3&-2\\6&6&1&6&6\\-&-&-+&+&+\\\end{array}}\\&=(-2\cdot 1)i+(4\cdot 6)j+(-3\cdot 6)k-(-3\cdot 1)j-(4\cdot 6)i-(-2\cdot 6)k\\&=-2i+24j-18k+3j-24i+12k\\&=-26i+27j-6k\\*|{\vec {c}}|&={\sqrt {(6-(-3))^{2}+(6-(-2))^{2}+(1-4)^{2}}}\\&={\sqrt {9^{2}+8^{2}+(-3)^{2}}}\\&={\sqrt {81+64+9}}\\&={\sqrt {81+64+9}}\\&={\sqrt {145}}\\*{\hat {b}}&={\frac {\vec {b}}{|{\vec {b}}|}}\\&={\frac {6,6,1}{\sqrt {6^{2}+6^{2}+1^{2}}}}\\&={\frac {6,6,1}{\sqrt {36+36+1}}}\\&={\frac {6,6,1}{\sqrt {73}}}\\*|{\vec {c}}|&={\frac {{\vec {a}}\cdot {\vec {b}}}{|{\vec {b}}|}}\\&={\frac {14}{\sqrt {73}}}\\*{\vec {c}}&={\frac {{\vec {a}}\cdot {\vec {b}}}{|{\vec {b}}|^{2}}}\cdot {\vec {b}}\\&={\frac {14}{({\sqrt {73}})^{2}}}(6,6,1)\\&={\frac {14}{73}}(6,6,1)\\\end{aligned}}}