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Unit Vectors

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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. 4.1 km 2.1 km 4.6 km

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Unit Length A vector with magnitude of exactly 1 has unit length. This is vector does not measure units like metersThis is vector does not measure units like meters Unit vectors have no units!Unit vectors have no units! The important feature of a unit vector is its directionThe 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. u = 1 A = 4.6 km

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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 x-direction. points in the y-direction points in the y-direction points in the z-direction points in the z-direction x y

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

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

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

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Unit Vector Notation 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 x = (2.0 km) cos(60°) = 1.0 km y = (2.0 km) sin(60°) = 1.7 km 60°

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

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