1. CHAPTER 3 VECTORS
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2. INTRODUCTION Definition 3.1
a VECTOR is a mathematical quantity that has both MAGNITUDE AND DIRECTION
VECTOR: represented by arrow where the direction of arrow indicates the DIRECTION of the vector & the length of arrow indicates the MAGNITUDE of the vector.
Eg: displacement, velocity, acceleration, force, ect
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3. INTRODUCTION Definition 3.2
a SCALAR is a mathematical quantity that has MAGNITUDE only
Scalar: represented by a single letter such as, k.
Eg: temperature, mass, length area, ect
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4. INTRODUCTION
Definition 3.3 A vector in the plane is a directed line segment that has initial point A and terminal point B, denoted by, ; its length is denoted by .
Initial Point, A
Terminal Point, B
Length:
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5. INTRODUCTION
Definition 3.5 Two vectors, and are said to be EQUAL if and only if they have the same MAGNITUDE AND DIRECTION.
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6. INTRODUCTION Definition : Component Form
If v is a 3-D vector equal to the vector with initial point at the origin and the terminal point , then the component form of v is defined by: z y x v1 v3 v2 O
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7. INTRODUCTION Definition : Magnitude / Length
the magnitude of the vector is:
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8. Example 1 Find : component form and
length of the vector with initial point P(-3,4,1) and terminal point Q(-5,2,2)
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9. VECTOR ALGEBRA OPERATIONS
Definition : Vector Addition and Multiplication by a Scalar
Let and be vectors with a scalar, k.
ADDITION
SCALAR MULTIPLICATION
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10. ADDITION OF VECTORS
The Triangle Law 2 vectors u and v represented by the line segment can be added by joining the initial point of vector v to the terminal point of u.
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11. ADDITION OF VECTORS
The Parallelogram Law The sum, called the resultant vector is the diagonal of the parallelogram. u v
u+v
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12. SUBTRACTION OF VECTORS
The subtraction of 2 vectors, u and v is defined by:
If and then,
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13. SUBTRACTION OF VECTORS
The subtraction of 2 vectors, u and v is defined by: u v
u+v -v
u-v
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14. THE SUM OF A NUMBER OF VECTORS
The sum of all vectors is given by the single vector joining the initial of the 1st vector to the terminal of the last vector. a b d c e
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15. SCALAR MULTIPLICATIONS OF VECTORS
Definition :
Let k be a scalar and u represent a vector, the scalar multiplication ku is:
A vector whose length |k| time of the length u and
A vector whose direction is:
The same as u if k>0 and
The opposite direction from u if k<0
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16. Example Let and . Find:
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17. PROPERTIES OF VECTOR OPERATIONS
Let u, v, w be vectors and a,b be scalars:
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18. UNIT VECTORS Definition
If u is a vector, then the unit vector in the direction of u is defined as:
A vector which have length equal to 1 is called a unit vector.
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19. DIRECTIONS OF ANGLES & DIRECTIONS OF COSINESz x y
- Are the angles that the vector OP makes with positive axis
- Known as the direction angles of vector OP P
DIRECTION OF COSINES O
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20. Example Find the direction cosines and direction angles of:
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21. DOT PRODUCT Also known as inner product or scalar product
The result is a SCALAR
If and then:
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22. Example If and :
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23. DOT PRODUCT Angle Between 2 Vectors
If the vectors lies on the same line or parallel to each other, then
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24. Example Find the angles between and
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25. Properties of Dot Product
if u and v are orthogonal
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26. CROSS PRODUCT
The result is a vector
If and then:
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27. Example 3 Find the cross product between and
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28. CROSS PRODUCT Properties of Cross Product if u and v are parallel
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