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1 Vectors and MatricesDefinition: mathematical expressions possessing a magnitude and a direction that add up accordingly to the parallelogram law.Examples: forces, displacements, velocities etc.

Graphical representation – an arrow with the direction of the vector and the length corresponding to the magnitude of the vector. Length of vector P is denoted as P or |P|.

Three basic types of vectors:Free vector is not attached to any point or a line in a body.Example: Displacement of the body moving without rotation.

Sliding vector has unique line of applicationExample: Force acting on a rigid body.

Fixed vector has unique point of application (and, thereof, line of application).Example: Force acting on a deformable body.

Note: in statics, forces are sliding vectors as there is no deformation. Hence, they have unique magnitude, direction, and line of action. However, we can slide the force along its line of action.

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Vector propertiesNegative vector

Multiplication by a scalar

a +(-a)=0

http://mathinsight.org/image/vector_opposite

http://webphysics.iupui.edu/JITTworkshop/152Basics/vectors/vectors.html

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Vector addition

Graphical Addition

Triangle rule:

Parallelogram rule:

Note 1: P+Q=Q+P

Note 2: The above methods do not general give line of action

Note 3: Parallelogram rule can be used to resolve the vector into components (not unique and not necessarily orthogonal)

Note 3: While P+Q=W , generally |P|+|Q|=|W|

http://www.maths.usyd.edu.au/u/MOW/vectors/vectors-3/v-3-3.html

http://www.maths.usyd.edu.au/u/MOW/vectors/vectors-3/v-3-3.html

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To find numerical values of length (magnitude) and angle (direction) of vector, use trigonometry.

Sine law A

sin (a )= Bsin (b )

= Csin (c ), good when two angles

and one side are known.

Cosine law C=√A2+B2−2 ABcos c, good for addition, when two vectors and angle between them are known.

Multiple vector addition

http://www.drcruzan.com/MathVectors.html

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Vector addition, Example 1:

Example 2

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Example 3

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Addition using rectangular components:2D

Cartesian unit vectors: |i|=|j|=1

Any vector can be then represented in Cartesian coordinates:

Q=Q x∗i+Q y∗ j

Where: Q x=|Q|∗cosΘ; Q y=|Q|∗sinΘ=|Q|∗cos(90−Θ)

From here:

|Q|=√Qx2+Q y

2 and Θ=tan−1Q y

Qx

Adding two vectors:Q=Q x∗i+Q y∗ j and P=P x∗i+Py∗ j

R=Q+P=R x∗i+Ry∗ j=Q x∗i+Q y∗ j+P x∗i+Py∗ j=(Q x+Px)∗i+(Q y+P y )∗ j

Similar for more than two forces.

Examples:

http://www.maths.usyd.edu.au/u/MOW/vectors/vectors-7/v-7-2.html

ϴ

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3D

Similar to 2D.

Unit vectors: |i|=|j|=|k|=1

Q=Q x i+Q y j+Q z k and |Q|=√Qx2+Q y

2+Qz2

Directional Cosines:If we have vector Q=Q x i+Q y j+Q z k, we can introduce directional cosines

cos (α )=Q x

Q,cos (β )=

Q y

Q,cos (γ )=

Q z

Q, where

α ,β , γ-coordinate direction angles between vector Q and positive i , j , k directions of coordinate system.

Then Q=Qcos (α ) i+Qcos (β) j+Qcos(γ)k

¿Q (cos (α ) i+cos (β ) j+cos (γ ) k ) ¿QuQ ,

Where uQ is unit vector in the direction of Q ,

uQ=cos (α ) i+cos (β ) j+cos (γ ) k and cos2 (α )+cos2 (β )+cos2 (γ )=1

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Matrices

Definition: A rectangular array of numbers arranged in rows and columns.

Matrix A of an order mXn

Each element of the matrix aij is the row i and column j element of A. For example, a12 denotes the element in the first row and second column.

Square Matrix: m=n

Note: A vector can be represented as a matrix of the order 1X3, in the following way:

Q=Q x i+Q y j+Q z k=(QxQ yQ z)

Submatrix is obtained from the matrix by deleting any collection of rows and columns.

Example:

Note: most of the matrix algebra is beyond the scope of this course.

http://en.wikipedia.org/wiki/Matrix_%28mathematics%29

http://en.wikipedia.org/wiki/Matrix_%28mathematics%29

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Determinant of a matrix:Determinant can be calculated only for a square matrix. Determinant of a matrix A is denoted det(A) or |A|.

For a 2X2 matrix, the determinant is defined as follows:

For matrices of higher order, the process is inductive. The determinant is calculated by going through either a row or a column, multiplying the elements in it by the determinant of a submatrix that is obtained by removing the row and the column of the corresponding element for the overall matrix and alternating the plus and minus signs. Of most interest to us is the determinant of a 3X3 matrix:

An alternative easy way to calculate the determinant of a 3X3 matrix is using the rule of Sarrus:

det (A )=a11a22 a33+a12a23a31+a13a21 a32−a13a22a31−a11a23 a32−a12 a21 a33

http://en.wikipedia.org/wiki/Determinant#2.C2.A0.C3.97.C2.A02_matrices

http://en.wikipedia.org/wiki/Determinant#2.C2.A0.C3.97.C2.A02_matrices

http://en.wikipedia.org/wiki/Determinant#2.C2.A0.C3.97.C2.A02_matrices

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Example of use: calculating the area of a parallelogram where the first row is 1’s and the second and third are vectors of the parallelogram’s sides.

Vector operatorsThe Scalar (dot) product:

Definition: A ∙B=|A|∙|B|∙cos (θ), where θ is the angle between vectors A and B, with their tails joined.

As can be seen the dot product is a scalar (hence the alternative name).

From definition it is obvious that for Cartesian unit vectors:i ∙ i= j ∙ j=k ∙ k=1 ∙1 ∙cos (0 )=1

i ∙ j=i ∙ k= j ∙ k=1 ∙1 ∙cos (90° )=0

Leading to:A ∙B=¿

Select computation method based on what is easier to compute or given.

Dot product rules

Commutative: A ∙B=B ∙ A

Distributive: A ∙ (B+C )=A ∙B+A ∙C

Multiplication by a scalar: a ( A ∙B )=(a A ) ∙B=A ∙(a B)

Example: A=2i−3 j, B=6 i+4 j, A ∙B=?.A ∙B=(2i−3 j ) ∙ (6 i+4 j )=2 ∙6−3 ∙4=0

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In order to calculate the angle between these two vectors:A ∙B=|A|∙|B|∙cos (θ )=0 ,⟹ϴ=90°

3D Example: A=i+3 j+k, B=2 i− j+k

A ∙B=?

A ∙B=1∗2−3∗1+1∗1=0

Hence, A and B are perpendicular. You can always check it by drawing them.

Applications of the dot product:

1) Angle between two vectors: θ=co s−1( A ∙BAB )You need to know components of A and B.

2) Components of vector parallel and perpendicular to a line.

F∥=F ∙cos (θ )=F ∙u∥

F ∥=F∥ ∙ u∥=(F ∙u∥) ∙ u∥

F⊥=F−F∥, F⊥=√F2−F ∥2

Alternatively: F⊥=F ∙ sin (θ ) ,

θ=co s−1( F ∙u∥F )F⊥=F⊥∙ u⊥=F ∙sin (θ ) ∙ u⊥

¿√F2−F∥2∙ u⊥

Examples:

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The vector (cross) product:

Definition: the vector product of two vectors A and B denoted as A×Bis defined as vector C which satisfies the following conditions:

1. Direction: perpendicular to the plane formed by A and B

2. Magnitude: |A×B|=|A|∗|B|∗sin(θ), where θ is an angle between |A| and |B|. Hence, A× A=0.

3. Sense: given by right hand rule (or screwdriver rule).

Note 1: From the product definition

|i ×i|=| j× j|=|k ×k|=1∙1∙ sin (0 )=0

|i × j|=|i×k|=| j×k|=1∙1∙ sin (90 ° )=1

Note 2: The vector product is not commutative A×B=−(B× A)

Note 3: The vector product is distributive

A× (B1+B2 )=A×B1+A×B2

How to calculate?

Example: using the definition

A=2i− j, B=3 i+2 j, A×B=?.

A×B= (2i− j )× (3 i+2 j )=2i × (3 i+2 j )− j× (3 i+2 j )=2 i×3 i+2 i×2 j− j×3 i− j×2 j=4k+3k=7k

Going by definition is cumbersome and complicated.

An alternative way to calculate:

It is easy to calculate both the magnitude and the direction of the cross product by using a determinant of a matrix.

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A×B=| i j kA x A y A z

B x By Bz|=( AyB z−A zB y )∗i+ (A zB x−A xBz )∗ j+( A xB y−A yBx )∗k

Mixed triple product: C ∙(A×B)

Interpretation: absolute value of the triple product is the volume of the parallelepiped with vectors A ,B∧C for sides.

Mixed triple product is zero if the vectors forming it are coplanar.

Counterclockwise circular permutation:

C ∙ (A ×B )=B ∙ (C × A )=A ∙ (B×C )=−C ∙ (B× A )=−A ∙ (C×B )=−B ∙(A ×C)

C ∙(A×B)=|C x C y C z

A x A y A z

B x By B z|

Note: the triple product is a scalar.

C

B

A