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Math 251 Final Review Pack

# Math 251 Final Review Pack - Chapter 12 Vectors 1 12.1...

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1 Chapter 12: Vectors

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2 12.1 Three-Dimensional Coordinate System 2D (Plane) A point is identified by an ordered pair (a,b) 3D (Space) A point is identified by an ordered triple (a,b,c,) *If a point P is identified by (a,b,c) we write: P(a,b,c) and (a,b,c) are called the coordinates of P Equations Surfaces ? Line Plane The distance between two points and : By the Pythagorean Theorem A sphere with radius r and center is of the form: (using the distance formula for 2 points) Regions of 3-D represented by equations or inequalities a) , a plane parallel to the xz- plane b) , everything to the right of the plane c) , everything bounded between the planes and including the planes d) , two planes of and e) , inside the sphere with radius and centered at (0,0,0) f) , cylinder
3 12.2 Vectors Vector: a geometrical quantity with both magnitude and direction Important: only relative position matters (as long as it has the same length and direction) Algebraic Operators: Addition: Given a vector u and v , we use the tip to tail method to define Scalar Multiplication: Given a vector v and a constant c such that , c v is a vector with magnitude multiplied by and its direction is the same as v if c >0 and opposite with c <0 Properties: 1) 2) 3) 4) 5) 6) 7) 8) Components Take a coordinate system and take a vector and place its tail on the origin and let ( a,b,c ) be the point that is on the head of the vector Warning: < > denotes a vector, ( ) denotes a point Magnitude of a Vector: Algebraic Operators: Addition, scalar multiplication and subtraction are done component-wise

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4 Special Vectors: These are the standard unit vectors of the standard basis: They are all of length 1
5 12.3 Dot Product Definition: Given 2 vectors and , the dot product is defined by: The dot product is a scalar not a vector Properties: 1) 2) 3) 4) 5) Geometric Interpretation: Where θ is the angel between a and b when they are drawn from the same initial point When θ is π/2 rad, the dot product of the vectors is 0 When the vectors are parallel θ=1 so: When the vectors are antiparallel θ= -1 so: Direction angles of a vector a : the angles that a make with the x, y and z axes and are denoted by α, β and γ Projections: The projection of a vector b onto a vector a, is defined as:

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6 12.3 The Cross Product Definition 1: Given 2 vectors and , the dot product is defined by: Definition 2: Given 2 vectors and , the dot product is denoted by: it is a vector that is orthogonal to a and b and its direction is determined by the right hand rule Theorem: Definitions 1 and 2 are equivalent Properties: 1) 2) , then a is parallel to b 3) 4) 5) 6) is the volume of the parallelepiped 7)
7 12.5 Equations of Lines and Planes A line is always determined when we know: a) The point on the line b) The direction vector of the line The parametric equations of the line are defined by: Where t is the parameter From this we can determine the symmetric equations of the line: When We can also determine the line segment from to using: A plane is determined always when we know: a) The point on the line b) The normal vector

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Math 251 Final Review Pack - Chapter 12 Vectors 1 12.1...

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