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Vectors TS: Explicitly assessing information and drawing conclusions.

Learn about vectors, directed line segments representing magnitude and direction. Explore vector operations, properties, unit vectors, and more. Practice examples to improve your skills in explicit information assessment and drawing conclusions.

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Vectors TS: Explicitly assessing information and drawing conclusions.

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  1. VectorsTS: Explicitly assessing information and drawing conclusions. Warm Up: What are the coordinates of A and B What is the distance between A and B?

  2. Vectors are directed line segments that are typically used to represent a force or speed (things that have both a magnitude and direction). The magnitude is indicated by the length of the vector, and the direction is indicated by the slope (or angle) of vector.

  3. If a plane is flying due North East at 100mph, then it’s motion can be represented as a vector. Which is shown as a terminating ray (or “directed segment”) 100mph N 45°

  4. If a person is pushing a box up a hill which has an angle of elevation of 15° using 16 Newtons (N) of force to do it, then it’s force can be shown using a vector, shown as this terminating ray (or “directed segment”) 16N 15°

  5. Equivalent Directed Line Segments A B C

  6. Component form of a Vector The component form of a vector with initial point P (p1, p2) and terminal point Q (q1, q2) is given by The magnitude of v is given by SPECIAL VECTORS: If ||v|| = 1, v is a unit vector. If ||v|| = 0, v is the zero vector. Q ||v|| v2 P v1

  7. Ex: Find the component form and magnitude of the vector shown, then drawn an equal vector whose terminal point is (5, 2)

  8. Vector Operations: Scalar Multiplication Given v find: • 2v • -v • 0.5v

  9. Vector Operations: Addition Given u and v Find u + v u v

  10. Vector Operations: Subtraction Find u - v u v

  11. Example: Multiple Operations Find u – 2v u v

  12. Properties of Vector Addition and Scalar Multiplication Let u, v, and w be vectors and let c and d be scalars. Then the following properties are true. • u + v = v + u • u + 0 = u • c(du) = (cd)u • c(u + v) = cu + cv • ||cv|| = |c| ||v|| • (u + v) + w = u + (v + w) • u + (-u) = 0 • (c + d)u = cu + du • 1(u) = u • 0(u) = 0

  13. Unit Vectors To find a unit vector divide the vector, v, by its magnitude. This will have the same direction as the vector, v, but it’s magnitude (or length) will be 1.

  14. Example: Find a unit vector going in the same direction as < -2, 5 >. Verify it is a unit vector.

  15. Writing a vector as a Linear Combination of Unit Vectors, & Unit Vectors: = <1, 0> and = <0, 1> Linear Combination of Unit Vectors v = < v1, v2> = v1 + v2

  16. Example: If u is a vector with initial side (2, -5) and terminal side (-1, 3), write u as a linear combination of the standard unit vectors and

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