Vector Algebra part 2
Descrição do Produto
6/28/2015
VECTOR MULTIPLICATION
VECTOR ALGEBRA
When two vectors A and B are multiplied, the result is either a scalar or a vector depending on how they are multiplied. 2 TYPES 1. Scalar (or dot) product: π β’ π 2. Vector (or cross) product: π Γ π
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VECTOR MULTIPLICATION
DOT PRODUCT
Multiplication of three vectors A, B, and C can result in either:
The dot product of two vectors A and B, written as π β’ π, is defined geometrically as the product of the magnitudes of A and B and the cosine of the angle between them.
1. Scalar triple product: π β’ (π Γ π) 2. Vector triple product: π Γ (π Γ π)
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DOT PRODUCT
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DOT PRODUCT If π = (π΄π₯ , π΄π¦ , π΄π§ ) and π = π΅π₯ , π΅π¦ , π΅π§
π β’ π = π΄π΅ cos ππ΄π΅ where: ππ΄π΅ is the smaller angle between A and B
π β’ π = π΄π₯ π΅π₯ + π΄π¦ π΅π¦ + π΄π§ π΅π§
The result of π β’ π is called either the scalar product because it is scalar, or the dot product due to the dot sign.
which is obtained by multiplying A and B component by component.
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DOT PRODUCT
DOT PRODUCT
Two vectors are said to be orthogonal (or perpendicular) with each other if πβ’π= 0
The dot product obeys the following: Commutative law:
πβ’π= πβ’π Distributive law:
πβ’ π+π = πβ’π+πβ’π π π β’ π = π = π΄2 ππ₯ β’ ππ¦ = ππ¦ β’ ππ§ = ππ§ β’ ππ₯ = 0 ππ₯ β’ ππ₯ = ππ¦ β’ ππ¦ = ππ§ β’ ππ§ = 1 12:46 PM
CROSS PRODUCT
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CROSS PRODUCT
The cross product of two vectors A and B, written as π Γ π, is a vector quantity whose magnitude is the area of the parallelopiped formed by A and B and is in the direction of advance of a righthanded screw as A is turned into B. 12:46 PM
CROSS PRODUCT
where:
The cross product of A and B is a vector with magnitude equal to the area of the parallelogram and the direction indicated. 12:46 PM
CROSS PRODUCT
π Γ π = π΄π΅ sin ππ΄π΅ ππ
ππ is a unit vector normal to the plane containing A and B The vector multiplication is called cross product due to the cross sign; it is also called vector product because the result is a vector.
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If π = (π΄π₯ , π΄π¦ , π΄π§ ) and π = π΅π₯ , π΅π¦ , π΅π§ ππ₯ π΄ πΓπ= π₯ π΅π₯
ππ¦ π΄π¦ π΅π¦
ππ§ π΄π§ π΅π§
= π΄π¦ π΅π§ β π΄π§ π΅π¦ ππ₯ + π΄π§ π΅π₯ β π΄π₯ π΅π§ ππ¦ +(π΄π₯ π΅π¦ β π΄π¦ π΅π₯ )ππ§ 12:46 PM
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CROSS PRODUCT
CROSS PRODUCT The cross product has the ff. basic properties: It is not commutative:
πΓπβ πΓπ It is anticommutative: π Γ π = βπ Γ π It is not associative: π Γ π Γ π β (π Γ π) Γ π It is distributive: Direction of π Γ π and ππ using (a) right-hand rule, (b) right-handed screw rule 12:46 PM
πΓ π+π = πΓπ+πΓπ πΓπ =π 12:46 PM
CROSS PRODUCT
SCALAR TRIPLE PRODUCT
The cross product has the ff. basic properties: ππ₯ Γ ππ¦ = ππ§ ππ¦ Γ ππ§ = ππ₯ ππ§ Γ ππ₯ = ππ¦
Given three vectors A, B, C, we define πβ’ πΓπ = πβ’ πΓπ = πβ’ πΓπ If π = (π΄π₯ , π΄π¦ , π΄π§ ) , π = π΅π₯ , π΅π¦ , π΅π§ and π = πΆπ₯ , πΆπ¦ , πΆπ§
Cross product using cyclic permutation: m(a) moving clockwise leads to positive results; (b) moving counterclockwise leads to negative results. 12:46 PM
Then π β’ π Γ π is the volume of a parallelepiped having A, B, and C as edges. 12:46 PM
SCALAR TRIPLE PRODUCT
VECTOR TRIPLE PRODUCT
By finding the determinant: π΄π₯ π΄π¦ π΄ π§ π β’ (π Γ π) = π΅π₯ π΅π¦ π΅π§ πΆπ₯ πΆπ¦ πΆπ§
For vectors A, B, and C, we define the vector triple product as πΓ πΓπ = π πβ’π βπ πβ’π
Since the result of this vector multiplication is scalar it is called scalar triple product.
Obtained using the βbac-cabβ rule. Note: π β’ π π β π(π β’ π) But π β’ π π = π(π β’ π)
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COMPONENTS OF A VECTOR
COMPONENTS OF A VECTOR
Vector product is used in determining the projection (or component) of a vector in a given direction.
Given a vector A, we define the scalar component π΄π΅ of A along vector B as π΄π΅ = π΄ cos ππ΄π΅ = A ππ΅ cos ππ΄π΅
π΄ π΅ = π β’ ππ©
The vector component π΄π΅ of A along vector B is simply the scalar component multiplied by a unit vector along B
ππ΅ = π΄π΅ β’ ππ© = (π β’ ππ© )ππ©
EXCERCISE 1.4
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If π = ππ₯ + 3ππ§ and π = 5ππ₯ + 2ππ¦ β 6ππ§ , find ππ΄π΅ .
120.6Λ 12:46 PM
EXAMPLE 1.5
COMPONENTS OF A VECTOR
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EXAMPLE 1.4
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Given vectors π = 3ππ₯ + 4ππ¦ + ππ§ and π = 2ππ¦ β 5ππ§ , find find the angle between A and B.
83.73Λ 12:46 PM
Three field quantities are given by π = 2ππ₯ β ππ§ π = 2ππ₯ β ππ¦ + 2ππ§ π = 2ππ₯ β 3ππ¦ + ππ§ Determine: a. (π + π) Γ (π β π) b. π β’ π Γ π c. π β’ π Γ π d. sin πππ
e. π Γ (π Γ π) f. A unit vector perpendicular to both Q and R g. The component of P along Q 12:46 PM
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Derive the cosine formula π2 = π 2 + π 2 β 2ππ cos π΄ and the sine formula sin π΄ sin π΅ sin πΆ = = π π π Using dot product and cross product respectively.
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Show that points π1 5, 2, β4 , π2 1, 1, 2 , and π3 β3, 0, 8 all lie on a straight line. Determine the shortest distance between the line and point π4 3, β1, 0
Yes, 2.426
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EXCERCISE 1.5
2ππ₯ + 12ππ¦ + 4ππ§ 14 14 0.5976 (2,3,4) Β± 0.745,0.298, β0.596 0.4444ππ₯ β 0.2222ππ¦ + 0.4444ππ§
EXCERCISE 1.6
a. b. c. d. e. f. g.
EXCERCISE 1.7
EXAMPLE 1.7
EXAMPLE 1.6
EXAMPLE 1.5
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Let π = 3ππ¦ + 4ππ§ and F= 4ππ₯ β 10ππ¦ + 5ππ§ (a) Find the component of E along F. (b)Determine a unit vector perpendicular to bot E and F.
(a) (β0.2837, 0.7092, β0.3546) (b) Β±(0.9398, 0.2734, β0.205)
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Show that vectors π = (4, 0, β1), π = (1, 3, 4), and π = β5, β3, β3 form the sides of a triangle. Is this a right triangle? Calculate the area of the triangle.
Yes, 10.5
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If π1 is 1, 2, β3 and π2 is β4, 0, 5 , find a) The distance π1 π2 b) The vector equation of the line π1 π2 c) The shortest distance between the line π1 π2 and point π3 (7, β1,2) a) 9.644 b) 1 β 5Ξ³ ππ± + 2(1 β Ξ³)ππ² + (8Ξ³ β 3)ππ³
c) 8.2
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