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Vector Geometry, Cheat Sheet of Mathematics

Norton University Mathematics

An overview of vector geometry, including the definition of vectors, vector operations such as addition, subtraction, and scalar multiplication, as well as the concepts of vector magnitude, direction, and orthogonality. It also covers the representation of vectors in a coordinate system, the calculation of vector dot products and cross products, and the applications of these concepts in areas like geometry and physics. Various examples and exercises to illustrate the key principles of vector geometry, making it a valuable resource for students and professionals working in fields that involve the use of vectors.

Typology: Cheat Sheet

2023/2024

Uploaded on 06/02/2024

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វ ៉ិចទ័រ

១.សញ្ញា ណវ ៉ិចទ័រ

វ ៉ិចទ័រជាអង្កត់ដែលមានទ៉ិសដៅ

ន៉ិង្អាំង្តង្់ស ៊ីដត (ម ៉ូឌ ល)។

𝐴𝐵

•𝐴𝐵ជាវ ៉ិចទ័រAB

•Aជាគល់, B ជាច ង្

•ទ៉ិសដៅដលើទាំរAB

•ទ៉ិសព៊ីAដៅB

•ម ៉ូឌ ល= ចមាា យ AB

២.ប្រមាណវ៉ិធ៊ីដលើវ ៉ិចទ័រ

ក.ទាំនាក់ទាំនង្សាល

𝐴𝐵𝑐

𝐵𝐶

𝐴𝐵

•𝑨𝑩+𝑩𝑪 = 𝑨𝑪

•𝑨𝑩+𝑩𝑪 + 𝑪𝑫+𝑫𝑬 + 𝑫𝑭= 𝑨𝑭

•𝑨𝑩+𝑩𝑪+ …… + 𝑴𝑵+……+ 𝑸𝑹= 𝑨𝑭

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𝐴𝐵 ជាវ ៉ិចទ័រ AB
A ជាគល់, B ជាច ង្
ទ៉ិសដៅដលើទាំរ AB
ទ៉ិសព៊ីAដៅB
ម ៉ូឌល = ចមាា យ AB

Q F

𝑨𝑩 + 𝑩𝑪 = 𝑨𝑪
𝑨𝑩 + 𝑩𝑪 + 𝑪𝑫 + 𝑫𝑬 + 𝑫𝑭 = 𝑨𝑭
𝑨𝑩 + 𝑩𝑪 + …… + 𝑴𝑵 +……+ 𝑸𝑹 = 𝑨𝑭

ខ.វ៉ិធានប្ រដលឡ៉ូប្ ាម

❖ រាំណកប្សាយ

ដយើង្ មាន 𝐴𝐵 & 𝐵𝐶
តាម A គ៉ូសរនាា ត់ប្សរ់ ន៉ិង្

𝐵𝐶

តាម B គ៉ូសរនាា ត់ប្សរ់ ន៉ិង្

𝐴𝐵

▪ រនាាត់ទាំព៊ីរាត់គ្ន ាប្ តង្់ D

▪ ដយើង្ បាន ប្ រដលឡ៉ូប្ ាម ABCD ដែល

𝑨𝑩 + 𝑩𝑪 = 𝑨𝑫 + 𝑫𝑪

𝑨𝑪 = 𝑨𝑪 ពិត

𝑨𝑩 + 𝑨𝑫 =? (𝟏)

ដោយ 𝑨𝑫 = 𝑩𝑪

( 1 ): 𝑨𝑩 + 𝑩𝑪 = 𝑨𝑪

⇒ 𝑨𝑫 + 𝑨𝑩 = 𝑨𝑪

+EX: តាមរ៉ូរខាង្ ដប្ ាមច៉ូសង្់ វ ៉ិចទ័រជាសលរ៉ូកររស់វា

𝐴𝐷 + 𝐴𝐵 = 𝐴𝐵 + 𝐵𝐶

= 𝐴𝐶

𝑣 Ԧ

𝑢 + 𝑣Ԧ =?

𝐴𝐵 + 𝑂𝐸 =?

𝐴𝐵 + 𝑂𝐸 = 𝑂𝐶 + 𝑂𝐸

= 𝐸𝐷 + 𝑂𝐸

= 𝑂𝐸 + 𝐸𝐷

= 𝑂𝐷

៣. ក៉ូអរដោដនននវ ៉ិចទ័រ

ដនាេះ a ∈ 𝑥

′

c ∈ 𝑧

′

𝑏

𝑎Ԧ

− 3

−∞

+∞

𝑎Ԧ = 2 = 2
𝑏 = − 3 = 3

𝑢 = (a, b, c)

𝑎 𝐷

𝑏

𝑥

𝑦

𝐸

𝑢

+ EX: ដៅចាំណច A ( 2 , - 3 , 4 ), B (- 2 , 1 , - 3 )

𝑜

𝑥

𝑦

𝐵

− 3

′

− 2

𝑍

𝐴

− 3

′

❖ ប្ ររ៉ិរតត៉ិ៖ ដគដអយ A (- 4 , 3 , - 2 )

B( 6 , - 2 , 5 ) & C( 2 , 1 , 4 )

𝑩

𝑨

𝑩

𝑨

𝑩

𝑨

៤. សាក ដលននព៊ីរវ ៉ិចទ័រ

′

𝟏

𝟐

𝟑

𝑼 ∙ 𝑽 = 𝟐 𝟒 + −𝟑 𝟑 + 𝟏 × 𝟏

𝑼 × 𝑽 ជាម ំដោែ

𝑼 × 𝑽 ជាម ំស្សួច

𝑼 × 𝑽 ជាម ំលក្ង

′

𝑼 × 𝑽 =

= 𝒃𝒄

′

𝒊Ԧ + 𝐜𝐚

′

𝒋Ԧ + 𝐚𝐛

′

𝒌 − (𝐛𝐚′𝒌 + 𝐂𝐛′𝒊Ԧ + 𝐚𝐜′𝒋Ԧ)

= 𝒃𝒄

′

− 𝒄𝒃

′

𝒊Ԧ − 𝒄𝒂

′

− 𝒂𝒄

′

𝒋Ԧ − (𝒂𝒃′ − 𝒃𝒂′)𝒊Ԧ

+ EX: 𝒂 𝟐, 𝟑, −𝟏 , 𝒃 −𝟏, 𝟐, −𝟐

គណនា 𝒂 ∙ 𝒃 េិង 𝒂 × 𝒃

𝒂 × 𝒃 =

𝒂 × 𝒃 = (−𝟒, 𝟓, 𝟕)

គ. 𝐴𝐵 × 𝐴𝐶, 𝐴𝐵 × 𝐵𝐶, 𝐴𝐶 × 𝐵𝐶

ចដមលើយ

ក.គណនា 𝐴𝐵, 𝐴𝐶, 𝐵𝐶

𝑨𝑩 = (𝒙

𝑩

− 𝒙

𝑨

) 𝒊Ԧ + 𝒚

𝑩

− 𝒚

𝑨

𝒋Ԧ + 𝒛

𝑩

− 𝒛

𝑨

𝒌

= (𝟎 − 𝟐) 𝒊Ԧ + 𝟑 − 𝟎 𝒋Ԧ + −𝟑 − 𝟏 𝒌

= −𝟐

Ԧ 𝒊 + 𝟑

Ԧ 𝒋 + 𝟒𝒌

ឬ 𝑨𝑩 −𝟐, 𝟑, −𝟒 , 𝑨𝑪(𝟎, 𝟓, −𝟏), 𝑩𝑪(𝟐, 𝟐, 𝟑)

ខ.គណនា 𝐴𝐵 ∙ 𝐴𝐶, 𝐴𝐵 ∙ 𝐵𝐶, 𝐴𝐶 ∙ 𝐵𝐶

𝑨𝑩 ∙ 𝑨𝑪 = −𝟐 𝟎 + 𝟑 𝟓 − 𝟒 −𝟏

= 𝟎 + 𝟏𝟓 + 𝟒 = 𝟏𝟗

𝑨𝑩 ∙ 𝑩𝑪 = 𝟐 −𝟐 + 𝟐 𝟑 + 𝟑(−𝟒)

= −𝟒 + 𝟔 − 𝟏𝟐 = −𝟏𝟎

𝑨𝑪 ∙ 𝑩𝑪 = 𝟎 𝟐 + 𝟓 𝟐 + 𝟑 −𝟏

= 𝟎 + 𝟏𝟎 − 𝟑 = 𝟕

គ.គណនា 𝐴𝐵 × 𝐴𝐶, 𝐴𝐵 × 𝐵𝐶, 𝐴𝐶 × 𝐵𝐶

𝐴𝐵 × 𝐴𝐶 =

𝒊 − 𝟐𝒋 − 𝟏𝟎𝒌ឬ 𝑨𝑩 × 𝑨𝑪(𝟏𝟕, −𝟐, −𝟏𝟎)

△𝑨𝑩𝑪

𝟐

𝟏

𝟐

△𝑨𝑩𝑪

𝟏

𝟐

𝐴𝐵 × 𝐴𝐶

𝑆

= 2 ∙

𝐴𝐵 × 𝐴𝐶

𝑆

= 𝐴𝐵 × 𝐴𝐶

𝐴𝐵𝐶𝐷

𝑨𝑩𝑪𝑫

𝐴

𝑧

𝐸 𝐹

𝐺

𝑦 𝐶

𝐷

𝑜

𝑥

𝐵

𝑉

𝑜𝐴𝐵𝐶𝐷𝐸𝐹𝐺= (𝑂𝐴×𝑂𝐶)∙𝑂𝐷

𝐴

𝐷

𝑂

𝐶

𝑉

(𝑂𝐴×𝑂𝐶)∙𝑂𝐷

△ = ቐ

𝑥 = 𝑥

𝑎𝑡

𝑦 = 𝑦

𝑏𝑡

𝑧 = 𝑧

𝑐𝑡

, (t ∈ ℝ)

0 ,

𝑜

𝑡 =

𝑜

, 𝑡 =

𝑜