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Engineering Calculus Notes 41

# Engineering Calculus Notes 41 - 1.2 VECTORS AND THEIR...

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1.2. VECTORS AND THEIR ARITHMETIC 29 where the (positive) scalar λ is λ = | −→ v | | −→ w | . Similarly, the two vectors point in opposite directions if the two unit vectors are negatives of each other, or −→ v = λ −→ w −→ w = 1 λ −→ v where the negative scalar λ is λ = | −→ v | | −→ w | . So we have shown Remark 1.2.1. For two nonzero vectors −→ v = ( x 1 ,y 1 ,z 1 ) and −→ w = ( x 2 ,y 2 ,z 2 ) , the following are equivalent: −→ v and −→ w point in parallel or opposite directions; −→ v = λ −→ w for some nonzero scalar λ ; −→ w = λ −→ v for some nonzero scalar λ ; x 1 x 2 = y 1 y 2 = z 1 z 2 = λ for some nonzero scalar λ (where if one of the entries of −→ w is zero, so is the corresponding entry of −→ v , and the
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