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Vectors

Vectors. What you should learn: Represent vectors as directed line segments. Write the component forms of vectors. Perform basic vector operations and represent them graphically. Write vectors as linear combinations of unit vectors. Find the direction angles of vectors.

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Vectors

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  1. Vectors What you should learn: Represent vectors as directed line segments. Write the component forms of vectors. Perform basic vector operations and represent them graphically. Write vectors as linear combinations of unit vectors. Find the direction angles of vectors. Use vectors to model and solve real-life problems.

  2. FYI Quantities that involve both magnitude and direction can’t be expressed by a single real number. Therefore, you need the concept of vectors. Force and velocity are examples of vectors.

  3. Vocabulary Vector – A line segment with both direction and magnitude. Magnitude – The length of a vector.

  4. Picture Representation of a Vector  Terminal point  Initial point

  5. FYI Two directed line segments that have the same magnitude and direction are equivalent.

  6. FYI Vectors are represented by lower case, boldface letters. Usually use the letters u, v and w

  7. Practice Let u be the vector from P(0,0) to Q(3,2). Let v be the vector from R(1,2) to S(4,4). Show that u and v are equivalent.

  8. Vocabulary Standard Position of a Vector- A vector whose initial point is the origin.

  9. Vocabulary Component Form of a Vector- Given initial point P(p1,p2) and terminal point Q(q1,q2), the component form would be… v = PQ = q1-p1,q2-p2

  10. Vocabulary Magintude of a Vector- The length of the vector can be found by using the Pythagorean Theorem/Distance Formula. v = (q1-p1)2 + (q2-p2)2

  11. Component Form of a Vector Vector Animation

  12. Component Form of a Vector x-component of a vector - drop a line from the tip of the original vector straight down to the x-axis and draw a vector along the x-axis from the origin to where this line hits the x-axis, then this newly drawn vector is the x-component of the original vector

  13. Component Form of a Vector y-component of a vector - drop a line from the tip of the original vector straight across to the y-axis and draw a vector along the y-axis from the origin to where this line hits the y-axis, then this newly drawn vector is the y-component of the original vector

  14. If v = 1, then v is called a unit vector. FYI If v = 0, then v is called the zero vector.

  15. Practice Find the component form and the magnitude of the vector, v, with initial point (-3,2) and terminal point (1,4). Interpret what your solution means.

  16. Component Form of a Vector What if I know the magnitude and the direction the vector is going. How can I find its component form? y M y M x M sin  = cos  =  x So x = Mcos  and y = Msin  Component form = Mcos  , Msin 

  17. Practice Find the component form of the vector that has a magnitude of 30 at an angle of 120.

  18. Vector Operations • Scalar Multiplication • Vector Addition

  19. Scalar Multiplication of Vectors Looks like… 2v or -3v or 3/4v What does it do? It changes the length and often times the direction of the vector.

  20. Scalar Multiplication Changed the length , but not the direction… the length was doubled. v 2v

  21. Scalar Multiplication Changed the length , but not the direction… the length was cut in half. v 1/2v

  22. Scalar Multiplication Changed the length , and the direction… the length multiplied by 2 and the direction reversed. v -2v

  23. Scalar Multiplication of Vectors How does it affect the component form? If v = u1,u2 then kv = ku1,u2 = ku1,ku2

  24. Addition of Vectors Looks like… u + v What does it do? It creates a resultant vector.

  25. Addition of Vectors u + v Position the two vectors you are trying to add without changing their lengths or directions. Rearrange the vectors so that the initial point of the second vector touches the terminal point of the first vector. Then create a vector by connecting the initial point of u to the terminal point of v. This is called the RESULTANT vector. u u + v v

  26. Example of Addition of Vectors v w u u + v + w

  27. Addition of Vectors How does it affect the component form? If u = x1,y1 and v = x2,y2 then u+v = x1+x2,y1+y2

  28. Addition of Vectors and Component Form

  29. Subtraction of Vectors Looks like… u - v What does it do? It creates a resultant vector. You need to think of it as… u + (- v)

  30. Subtraction of Vectors u – v or u + (-v) u Then create a vector by connecting the initial point of u to the terminal point of -v. This is called the RESULTANT vector. Rearrange the vectors so that the initial point of the second vector touches the terminal point of the first vector. Position the two vectors you are trying to subtract without changing their lengths or directions. Think what negative v would look like. v -v

  31. Subtraction of Vectors How does it affect the component form? If u = x1,y1 and v = x2,y2 then u-v = x1-x2,y1-y2

  32. Summary of Vectors • There exists a zero vector. • A vector of magnitude one is called a unit vector. • A vector A multiplied by a scalar m is a vector, unchanged in direction, but modified in length by the factor m. • The negative of a vector is the original vector flipped 180 degrees. • Two vectors, A and B, are added by placing the tail of one on the tip of the other (in either order) and defining the sum to be the vector drawn. from the tail of the first to the tip of the second. • A vector B can be subtracted from a vector A by adding -B to A.

  33. Practice Given: u = 3,4 ; v = 6,-2 ; w = -2,1 Find: u + v 3v –2w w – u +4v

  34. Dot Product of Vectors The Dot Product of Vectors does not result in another vector. Instead, it results in a scalar value (a single number). If u = x1,y1 and v = x2,y2 then uv = x1x2+y1y2

  35. Practice Given: u = 3,4 ; v = 6,-2 ; w = -2,1 Find: w v  w u  v + u

  36. Dot Product of Vectors Where is this useful? It is needed to find the angle between two vectors.

  37. Angle between 2 Vectors cos  = v u v u

  38. If v = 1, then v is called a unit vector. Unit Vectors Two Basic Unit Vectors i = 1,0 which is 1 unit long and rests along the x axis. j = 0,1 which is 1 unit long and rests along the y axis.

  39. Unit Vectors To find a unit vector with the same direction as a given vector, we divide by the magnitude of the vector. For example, consider the vectorv = 1, 3 which has a magnitude of . If we divide each component of v by we will get the unit vectoruv which is in the same direction as v.

  40. Practice Given: u = 3,4 ; v = 6,-2 ; w = -2,1 Find: A unit vector that is in the same direction as w. (uw) Describe v using the basic unit vectors, i & j.

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