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Section 6.3: Vectors in a Plane

Section 6.3: Vectors in a Plane. Mesa Verde National Park, Colorado. Greg Kelly, Hanford High School, Richland, Washington. Photo by Vickie Kelly, 2003. Edited by: JHeyd. Quantities that we measure that have magnitude but not direction are called scalars.

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Section 6.3: Vectors in a Plane

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  1. Section 6.3: Vectors in a Plane Mesa Verde National Park, Colorado Greg Kelly, Hanford High School, Richland, Washington Photo by Vickie Kelly, 2003 Edited by: JHeyd

  2. Quantities that we measure that have magnitude but not direction are called scalars. Quantities such as force, displacement or velocity that have direction as well as magnitude are represented by directed line segments. B terminal point The length is A initial point

  3. B terminal point A initial point A vector is represented by a directed line segment. Vectors are equal if they have the same length and direction (same slope).

  4. y A vector is in standard position if the initial point is at the origin. x The component form of this vector is:

  5. y A vector is in standard position if the initial point is at the origin. x The component form of this vector is: The magnitude (length) of is:

  6. The component form of (-3,4) P is: (-5,2) Q v (-2,-2)

  7. Then v is a unit vector. If is the zero vector and has no direction.

  8. Vector Operations: (Add the components.) (Subtract the components.)

  9. Vector Operations: Scalar Multiplication: Negative (opposite):

  10. Sum of Two Vectors u v u + v is the resultant vector. u+v (Parallelogram law of addition) v u

  11. Difference of Two Vectors v u – v u u – v v

  12. Unit Vectors: Note that u a scalar multiple of v. The vector u is called a unit vector in the direction of v.

  13. The unit vectors and are called standard unit vectors and are denoted by and . These vectors can be used to represent any vector as follows: The scalars are called the horizontal and vertical components of v. The vector sumis called a linear combination of the vectors I and j.

  14. Any vector can be written as a linear combination of two standard unit vectors. The vector v is a linear combination of the vectors i and j. The scalar a is the horizontal component of v and the scalar b is the vertical component of v.

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