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Understanding Vectors & Their Components

Learn about scalar and vector quantities, notations, forms of vectors, adding vectors geometrically and in written form, unit vectors, and more.

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Understanding Vectors & Their Components

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  1. VECTORS Dr. Shildneck

  2. Vectors There are two types of quantities in the world. • Scalar – a quantity that is specified by a single value with an appropriate unit and has no direction. (Examples: temperature, height, length of string) • Vector – a quantity that has both magnitude (size) and direction. (Examples: driving west at 50 mph, pulling a cart up a hill, weight)

  3. Vectors Q Example: The vector from P to Q. Vectors have non-negative magnitude (size) and a specific direction. To represent them, we use directed line segments. The segments have an initial point and a terminal point. At the terminal point, we represent the direction of the vector with an “arrow head.” P

  4. Notations Notations for Vectors • using points: (use a half arrow over the points in order) • using vector name (typed): v(bold lowercase letter) • using vector name (handwritten): (lowercase letter with half arrow) Notation for Magnitude

  5. Vectors Vectors are said to be in “Standard Position” if its initial point is (re-)located at the origin. Most vectors can be thought of as position vectors because any vector can be re-positioned at the origin. By initiating a vector at the origin, we can easily manipulate the horizontal and vertical components of the vector.

  6. Component Form The component form of a vector is written as the “end point” when the vector is in standard position (initial point at the origin). The component form of the vector from P(a, b) to Q(c, d) can be found by subtracting the components of each point. <c-a, d-b> This is also called the position vector.

  7. Linear Combinations The linear combination form of a vector uses scalars of the standard unit vectors i = <1, 0> and j = <0, 1> to write the position vector. For example, the vector <3, 5> can be written in the form 3i + 5j Because 3i + 5j = 3<1, 0> + 5<0, 1> = <3, 0> + <0, 5> = <3, 5>

  8. Forms of Vectors Write the vector from P to Q in both its component form and as a linear combination of i and j. P = (3, 8) Q = (-4, 18) In general, in what direction does this vector head?

  9. Forms of Vectors Q Up 8 or +8 Direction: Left 5 and Up 8 or Up & Left with a slope of -8/5 P Left 5 or -5 Find the magnitude and describe the actual direction of the vector from P to Q. P = (3, -4), Q = (-2, 4)

  10. Equal Vectors Vectors are equal if they have the same magnitude AND direction. Location does not matter when determining if vectors are equal. To show that two vectors are equal, show that their magnitude is the same and that they travel in exactly the same direction.

  11. Equal Vectors Determine if each of the following pairs of vectors are equal. If not, state why not. Initial PointTerminal Point A(2, 2) B(5, 7) R(4, 3) S(7, 8) A(3, 1) B(-2, -7) R(1, -4) S(6, 4)

  12. Scalar Multiplication – Resizing Vectors Any vector can be resized by multiplying it by a real number (scalar). Multiplying by positive scalar changes magnitude only. Multiplying by a negative scalar changes the magnitude and its direction.

  13. Resizing Written Vectors “distribute” 4u = 4<3, 5> = <4x3, 4x5> = <12, 20> Example: Given u = <3, 5>, find 4u.

  14. Adding Vectors – Geometrically “Parallelogram Method” Given two vectors, to add them geometrically, you can use a parallelogram. First, join the vectors initial points (tails). Second, create two more vectors that are equal to the original vectors. Place them where the tails meet the heads of the first set and join their heads to make a parallelogram. Finally, the resultant vector of this addition is the diagonal from the joined tails to the joined heads.

  15. Adding Vectors – Geometrically Join the tails of the two vectors you are adding. Create two equal vectors. Join the new vectors’ tails to the heads of the original. Draw the diagonal from the TAIL to the HEAD. u + v u v “Parallelogram Method”

  16. Adding Vectors in Written Form Adding vectors in written forms is fairly simple. Basically you just have to follow the order of operations. In component form: Multiply through by any scalars. Add horizontal components, Add vertical components In Linear combinations: Combine like terms.

  17. Adding Vectors in Written Form 3<3, 5> + 2 <2, -4> = <9, 15> + <4, -8> = <13, 7> 2<3, 5> - <2, -4> = <6, 10> + <-2, 4> = <4, 14> Examples. Given u = <3, 5> and v = <2, -4> find the following vectors. 3u + 2v = 2u - v =

  18. Unit Vectors A unit vectoris a vector of magnitude 1 (in any direction). To find a unit vector in a specific direction (the direction of another given vector), you must “divide” the given vector using scalar multiplication so that the new vector’s magnitude is 1. • Find the magnitude of the given directional vector. • Multiply by the reciprocal of the magnitude.

  19. Unit Vectors 1. Find the magnitude of <-3, 4> ||<-3, 4>|| 2. Find the unit vector by multiplying by the reciprocal. 3. Find the unit vector in linear combination form. Example: Find the unit vector in the same direction as <-3, 4>.

  20. Unit Vectors 1. Find the Unit Vector. 2. Multiply by the needed magnitude. Example: Find the vector of magnitude 30 in the same direction as <-3, 4>.

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