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Introduction to Vectors UNIT 1
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What is a Vector? A vector is a directed line segment, can be represented as AB or AB, where A and B are the two endpoints of the line segment. Directed means that the vector has a direction. QUESTION: Which direction is implied for vector AB?
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Vectors: Example A B
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Vector Quantity There are exactly two properties that completely characterize a vector: 1)Direction – which way does the vector point? 2)Magnitude – the length of the vector, written as |XY| for vector XY Together the direction and magnitude define the vector quantity. QUESTION: What are some examples of vectors that we are already familiar with?
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Equal Vectors Two vectors are equal vectors if they have both the same direction and magnitude
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Three cars on the road are driving in the same direction at the same speed: Do they have equal velocity vectors?
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Opposite Vectors: An opposite vector is a vector with the same magnitude as the original, but opposite direction: A B In this illustration, AB and BA are opposite vectors.
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Vector Sum A vector sum A + B is defined as a vector that results from placing the initial point of vector B at the terminal point of vector A: the vector with the same initial point as A and the same terminal point as B is the vector sum. A B A + B
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Parallelogram Rule Another method, called the parallelogram rule, to find A + B is to place vectors A and B so that their initial points coincide, then complete a parallelogram that has A and B as two adjacent sides. The diagonal of the parallelogram with the same initial points is the vector sum:
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Parallelogram Rule Example A B A + B
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The two methods side by side: A B A + B A B The two methods give identical results for the vector sum
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QUESTION Two forces of 15 newtons and 22 newtons act at a point in the plane. If the angle between the forces is 100°, find the magnitude of the resultant force: 22 15 100°
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QUESTION Two forces of 15 newtons and 22 newtons act at a point in the plane. If the angle between the forces is 100°, find the magnitude of the resultant force: 22 15 =80 Z 22 100°
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Scalar Product A real number k and a vector U create the vector kU which has the magnitude |k| times the magnitude of U. kU has the same direction as U if k>0, and the opposite direction if k < 0: – So vector 2U would be twice the length of U and would point in the same direction as U does:
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Scalar Product Examples U 2U 3U -U -2U -3U
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Position Vectors A position vector is a vector with its initial point at the origin and with its endpoint at (a, b). It is written, so U = below: (a, b) U X axis Y axis (0, 0)
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Direction Angle of The direction angle is the positive angle between the x-axis and a position vector: (a, b) U X axis Y axis (0, 0) Direction Angle
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Direction Angle of (cont) (a, b) U X axis Y axis (0, 0) The direction angle θ satisfies tan θ = b/a, where a ≠ 0: b a
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Magnitude of vector (a, b) U X axis Y axis (0, 0) b a
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QUESTION What is the direction angle and magnitude of vector U= ? (3, -2) U X axis Y axis (0, 0)
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QUESTION
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Horizontal and Vertical Components of a Vector The horizontal and vertical components of a vector U are given by: a = |U|cos θ b = |U|sin θ (a, b) U X axis Y axis (0, 0) b = |U|sin θ a = |U|cos θ
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QUESTION Calculate the vertical and horizontal components of a vector with direction angle of 40° and a magnitude of 25. (a, b) |U|=25 X axis Y axis (0, 0) = 40
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QUESTION Calculate the horizontal and vertical components of a vector with direction angle of 40° and a magnitude of 25. ANSWER: x = 19.2, y = 16.1 or
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Vector Operations Overview + = k* = If A =, then –A = - = + - OR + =
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Vector Operations For any real numbers a, b, c, and d: + = (a, b) X axis Y axis (0, 0) (c, d) (a+c, b+d)
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Vector Operations For any real numbers a, b, c, and d: + = (3, 4) X axis Y axis (0, 0) (4, -2) (7, 2)
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Vector Operations (cont) B If U = Then –U =
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Vector Operations Scalar multiplication: k* = Examples: -3* = 6* = 0* =
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Vector Operations (0, 0) (3, 4) (0, 0) (6, 8) U 2U
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QUESTION Consider the vectors shown in the following figure, and perform the operations: U V U + V (4, 3) X axis Y axis (0, 0) (-2, 1) (x, y) a)U + V b)-2U c)4U – 3V
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Vector Subtraction A B -A -B Vector subtraction is the inverse operation of vector addition and is defined as adding the negative vector: So we have B – A = B + (-A) for all vectors A, B Therefore (see below) B – A = C C
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Vector Subtraction - = + - OR + = Examples: - =
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Vector Subtraction QUESTIONS QUESTION: Express A as a difference of two vectors. QUESTION: Express B as a difference of two vectors. A B C
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Vector Notation Conventions Unit Vectors: i =, j= i, j Form for Vectors: If v =, then v = ai + bj
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Unit Vectors (5, 3) U X axis Y axis (0, 0) i i i i i j j j i j U= 5i + 3j
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QUESTION Write the vector in the form ai + bj:
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QUESTION Write the vector in the form ai + bj: Answer: If v=, then: V=ai + bj, so V = -5i + 8j
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Dot Product The dot product of the two vectors U = and V = is denoted by UV, read “U dot V,” is given by UV = ac + bd. Examples: =
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Geometric Interpretation of the Dot Product If θ is the angle between two nonzero vectors U and V, where 0° < θ < 180°, then UV = |U|*|V|cos θ Example: = 6 using =ac+bd But it is also true using UV = |U|*|V|cos θ
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Geometric Interpretation of the Dot Product (3, 3) U X axis Y axis (0, 0) (2, 0) V θ=45
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Properties of the Dot Product UV = VU U(V+W)=UV + UW (U + V)W = UW + VW (kU)V = k(UV) = U(kV) 0U = 0 UU = |U| 2
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The Dot Product Can Be Positive, Zero, or Negative θ θ < 90: Positive dot product θ θ θ > 90: Negative dot product θ = 90: Zero dot product
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