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Vectors

Vectors. Geometric and Algebraic. Geometric Vectors. A vector is a quantity that has both magnitude and direction. We use an arrow to represent a vector. The length of the arrow is the magnitude and the arrowhead indicates the direction of the vector. Geometric Vectors.

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Vectors

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  1. Vectors Geometric and Algebraic

  2. Geometric Vectors • A vector is a quantity that has both magnitude and direction. • We use an arrow to represent a vector. • The length of the arrow is the magnitude and the arrowhead indicates the direction of the vector.

  3. Geometric Vectors • The points on a line joining the two points form what is called a line segment. • If we order the points so that they proceed from one point to another, we have a directed line segment or a geometric vector, denoted by the two points with an arrow on top of it. • The beginning point is called the initial point. • The ending point is called the terminal point.

  4. Zero Vector • The vector v whose magnitude is 0 is called the zero vector, 0. The zero vector is assigned no direction. • Two vectors are equal if they have the same magnitude and direction. (Both must be the same.) • See example on p. 619.

  5. Adding Vectors • To find the sum of two vectors we position the vectors so that the terminal point of the first vector coincides with the initial point of the second vector. • Vector addition is commutative • v + w = w + v Vector addition is also associative u + (v + w) = (u + v) + w

  6. Vector Addition • The zero vector has the property that • v + 0 = 0 + v = v • If v is a vector, then –v is the vector having the same magnitude as v, but whose direction is opposite to v. • v + (-v) = 0

  7. Multiplying Vectors by Numbers • If a is a scalar and v is a vector, the scalar product av is defined as follows: • If a > 0, the product av is the vector whose magnitude is atimes the magnitude of v and whose direction is the same as v. • If a<0, the product av is the vector whose magnitude is |a|times the magnitude of v and whose direction is the opposite that of v. • If a = 0 or if v = 0, then av = 0.

  8. Scalar Multiplication Properties • 0v = 0 1v = v -1v = -v • (a + b)v = av + bv (a + b)v = av + bv • a(bv)=(ab)v

  9. Graphing Vectors • p. 628 #s 2 – 8 even

  10. Magnitudes of Vectors • If v is a vector, we use the symbol ||v|| to represent the magnitude. ||v|| is the length of a directed line segment.

  11. Properties of Vectors • If v is a vector and if a is a scalar, then • (a) ||v|| ≥ 0 (b) ||v|| = 0 iff v = 0 • (c) ||-v|| = ||v|| (d) ||av|| = |a| ||v||

  12. Unit Vector • A vector u for which the magnitude of vector u is equal to one is called a unit vector.

  13. Algebraic Vectors • An algebraic vector is represented by an ordered pair written in brackets. where a and b are real numbers (scalars) called the components of the vector v.

  14. Position Vector

  15. Finding a Position Vector • Find the position vector of the vector

  16. Equality of Vectors • Two vectors v and w are equal iff their corresponding components are equal. We call a and b the horizontal and vertical components of v, respectively.

  17. Unit Vectors

  18. Addition and Subtraction Using Unit Vectors

  19. Scalar Product and Magnitude Using Unit Vectors

  20. Unit Vector in the Direction of v • For any nonzero vector v, the vector • is a unit vector that has the same direction as v.

  21. Finding a Unit Vector • Find a unit vector in the same direction as • v = 2i – j • We find ||v|| first.

  22. Writing a Vector in Terms of Its Magnitude and Direction • If a vector represents the speed and direction of an object, it is called a velocity vector. • If a vector represents the direction and amount of a force acting on an object, it is called a force vector. • The following helps to find the velocity and force vectors.

  23. Vector in Terms of Magnitude and Direction • A vector written in terms of magnitude and direction is • v = ||v|| (cos ai + sin aj) • where a is the angle between v and i.

  24. Writing a Vector when Magnitude and Direction Are Given • A child pulls a wagon with a force of 40 pounds. The handle of the wagon makes an angle of 30o with the ground. Express the force vector F in terms of i and j. • The magnitude of the Force is given as 40 pounds. • Now express the vector in terms of magnitude and direction

  25. Force Problem Continued

  26. Static Equilibrium • If two forces simultaneously act on an object the vector sum is known as the resultant force. An application of this concept is static equilibrium. An object is said to be in static equilibrium if (1) the object is at rest and (2) the sum of all forces acting on the object is zero.

  27. Object in Static Equilibrium • A weight of 1000 pounds is suspended from two cables as shown in the figure on p. 629 problem 59. What is the tension of the two cables.

  28. Object in Static Equilibrium • On-line Example

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