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)
Answer a) b)
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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. Initial point of v/ Terminal point of u
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.
Initial point of v/ Terminal point of u
Terminal point of u
Initial 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: Answer a) b) c)
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17. PARALLEL VECTORS
Definition : If two vectors, u and v have the same direction, whether their magnitudes are same or not, they are said to be parallel. To be parallel vectors, one should a scalar multiple of another.
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18. PROPERTIES OF VECTOR OPERATIONS
Let u, v, w be vectors and a,b be scalars:
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19. 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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20. 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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21. Example Find the direction cosines and direction angles of: Answer (i) (ii)
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22. DOT PRODUCT Also known as inner product or scalar product
The result is a SCALAR
If and then:
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23. Example
If and
Answer 9
-18
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24. DOT PRODUCT Angle Between 2 Vectors
If the vectors lies on the same line or parallel to each other, then
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25. PERPENDICULAR (ORTHOGONAL) VECTORS
Two vectors are perpendicular or orthogonal if the angle between them is Definition Vectors u and v are perpendicular (orthogonal) if and only if u.v =0.
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26. Example Find the angles between and Answer (i) (ii)
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27. Properties of Dot Product
if u and v are orthogonal
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28. CROSS PRODUCT
The result is a vector If and then:
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29. CROSS PRODUCT
The vector is orthogonal to both u and v also known as the normal vector, n.
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30. Example 3 Find the cross product between and Answer (i) (ii)
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31. CROSS PRODUCT Properties of Cross Product if u and v are parallel
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32. VECTORS
APPLICATIONS
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33. Line L is the set of all points P(x,y,z) for which parallel to :
APPLICATIONS
Lines & Line Segment in Space
Parametric Equations z y x
P0(x0,y0,z0) L
P(x,y,z) v
Line L is the set of all points P(x,y,z) for which parallel to :
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34. APPLICATIONS
Therefore the parametric equation for L : Cartesian equation:
Parametric
Equation
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35. Example 1 Find parametric and Cartesian equation for the line passes through Q(-2,0,4) and parallel to Answer Parametric Equation Or Cartesian Equation
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36. Example 2 Find the parametric equation for the line passes through P(-3,2,-3) and Q(1,-1,4) Answer At point P At point Q
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37. APPLICATIONS Distance from point S to line L Lv L
From the properties of Cross Product
Formula of Distance from point S to L
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38. Example 3 Find the distance from the point S(1,1,5) to the line Solution
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39. APPLICATIONS Equation of Planes
Vector is on the plane M and vector which is perpendicular to M known as normal vector, n
Equation of Plane is defined by:
From the properties of Dot Product
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40. APPLICATIONS
Equation of Planes
Normal vector n :
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41. APPLICATIONS
Let and
EQUATION OF PLANE
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42. Example 4 Find an equation of plane through P0(-3,0,7) perpendicular to Answer
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43. Example 7: Find equation of plane through 3 points: Solution Find the normal vector, n: Equation of plane: (use any point A,B or C)
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44. APPLICATIONS
Parallel Planes 2 planes parallel if and only if their normal planes are parallel
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45. APPLICATIONS
Lines of Intersection 2 planes that are not parallel, intersect in a line.
Normal vector for plane M and N
Line of intersection
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46. APPLICATIONS
Lines of Intersection From the properties of cross product, the result between vector product is a vector. Therefore, whenever two vectors crossing with each other, new vector will be produced.
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47. Example 8: Find a vector parallel to the line of intersection of the planes Answer
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48. Example 9: Find the parametric equation for the line in which the planes intersect. Solution Find v
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49. Find the intersection point: If x =0, Therefore, the point of intersection is The parametric equation for and point is:
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50. Example 9: Find the point where the line Intersects the plane Solution Find t Therefore, point of intersection is
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51. APPLICATIONS Distance from a Point to the Plane P n D P0
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52. APPLICATIONS
Equation of plane
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53. Example 10: Find the distance from S(1,1,3) to the plane Answer
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