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Understanding Vector Operations: Magnitude, Direction, and Linear Combinations

This lesson explores fundamental vector operations, including how to find distances between points, calculate vector magnitudes, and express vectors in component form. We'll learn about equivalent vectors, scalar multiplication, vector addition, and subtraction. The concept of unit vectors and how to express any vector as a linear combination of unit vectors is emphasized. Students will also discover how to determine direction angles for vectors. Exercises and examples will solidify understanding and application of vector operations.

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Understanding Vector Operations: Magnitude, Direction, and Linear Combinations

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  1. 7.6 Vector OperationsDay 1 Do Now Find the distance between the points (-4, -3) and (1, 5)

  2. Vector Notation • A vector V whose initial point is the origin and whose terminal point is (a, b) can be written as where a is the horizontal component and b is the vertical component of the vector

  3. Component Form of a Vector • The component form of a ray AC with and is

  4. Ex • Find the component form of the ray CF if C = (-4, -3) and F = (1, 5)

  5. Length of a Vector • The length, or magnitude, of a vector is given by

  6. Ex • Find the magnitude of vector

  7. Equivalent Vectors • Vectors are equivalent if they have the same magnitude and direction • This means they don’t have to start/end in the same position

  8. Vector Operations • Scalar Multiplication • Vector Addition • Vector Subtraction • Zero Vector

  9. Ex • Evaluate the following, where • 1) u + v • 2) u – 6v • 3) 3u + 4v • 4) |5v – 2u|

  10. Unit Vector • A vector of magnitude 1 is called a unit vector • If v is a vector and is not a zero vector, then a unit vector with the same direction as v is

  11. Ex • Find a unit vector that has the same direction as the vector

  12. Linear Combinations • The unit vectors parallel to the x and y axes are defined as • Any vector can be expressed as a linear combination of unit vectors I and j

  13. Ex • Express vector as a linear combination of I and j

  14. Ex • Write the vector q = -I + 7j in component form

  15. Ex • If a = 5i – 2j and b = -I + 8j, find 3a – b in component form

  16. Direction Angle • The direction angle of a vector is the angle of the triangle created by the vector and the positive half of the x-axis

  17. Ex • Find the direction angle of the vector w = -4i – 3j

  18. Closure • Determine the direction angle of the vector • HW: p.666 #1-57 odds

  19. 7.6 Vector OperationsDay 2 • Do Now • Given the vector • 1) Find the magnitude of u • 2) Find the unit vector that has the same direction • 3) Express the vector as a linear combination of I and j

  20. HW Review: p.666 #1-57 odds

  21. Closure • None • HW: none

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