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Engineering Calculus Notes 37

Engineering Calculus Notes 37 - 25 1.2 VECTORS AND THEIR...

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1.2. VECTORS AND THEIR ARITHMETIC 25 −→ v −→ w −→ u −→ v - −→ w −→ u −→ u = −→ v + −→ w −→ v = −→ u −→ w Figure 1.20: Difference of vectors associative: −→ u + ( −→ v + −→ w ) = ( −→ u + −→ v ) + −→ w . Multiplication of vectors by scalars distributes over vector sums: r ( −→ v + −→ w ) = r −→ w + r −→ v , and distributes over scalar sums: ( r + s ) −→ v = r −→ v + s −→ v . We will explore some of these properties further in Exercise 3 . The interpretation of displacements as vectors gives us an alternative way to represent vectors. We will say that an arrow representing the vector −→ v is in standard position if its tail is at the origin. Note that in this case the vector is completely determined by the position of its head, giving us a natural correspondence between vectors −→ v in R 3 (or R 2 ) and points P R 3 ( resp . R 2 ). −→ v corresponds to P if the arrow −−→ O P from the origin to P is a representation of −→ v : that is, −→ v is the vector representing that displacement of R 3 which moves the origin to
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