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Unit Vectors

Unit Vectors. Vector Length. Vector components can be used to determine the magnitude of a vector. The square of the length of the vector is the sum of the squares of the components. 4.6 km. 2.1 km. 4.1 km. Unit Length. A vector with magnitude of exactly 1 has unit length .

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Unit Vectors

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  1. Unit Vectors

  2. Vector Length • Vector components can be used to determine the magnitude of a vector. • The square of the length of the vector is the sum of the squares of the components. 4.6 km 2.1 km 4.1 km

  3. Unit Length • A vector with magnitude of exactly 1 has unit length. • This is vector does not measure units like meters • Unit vectors have no units! • The important feature of a unit vector is its direction • A vector can be made by multiplying a scalar magnitude times a unit vector in the proper direction. A = 4.6 km u = 1

  4. Cartesian Coordinates • A special set of unit vectors are those that point in the direction of the coordinate axes. • points in the x-direction. • points in the y-direction • points in the z-direction y x

  5. Unit Vectors or Components • A vector can be listed in components. • A vector’s components can be used with unit vectors.

  6. y x Projection • A vector is projected onto each coordinate axis. • The magnitude of the projection is multiplied times a unit vector. q

  7. Projection and Trigonometry • The use of trigonometry can be combined with the projections onto the coordinate axes. • The magnitude of A and the angle q become components. • The vector A is represented by the components and unit vectors.

  8. Write the vector of magnitude 2.0 km at 60° up from the x-axis in unit vector notation Find the components. x = r cos  = 1.0 km y = r sin  = 1.7 km Use unit vectors y x Unit Vector Notation y = (2.0 km) sin(60°) = 1.7 km 60° x = (2.0 km) cos(60°) = 1.0 km

  9. Alternate Axes • Projection works on other choices for the coordinate axes. • Other axes may make more sense for the physics problem. y’ x’ f next

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