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Engineering Calculus Notes 68

# Engineering Calculus Notes 68 - −→ w | 2 iii Show that...

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56 CHAPTER 1. COORDINATES AND VECTORS Place the parallelogram with one vertex at the origin: suppose the two adjacent vertices are P and Q and the opposite vertex is R (Figure 1.31 ). Represent the sides by the vectors −→ v = −−→ O P = −−→ QR −→ w = −−→ O Q = −→ PR. i. Show that the diagonals are represented by −−→ O R = −→ v + −→ w −−→ PQ = −→ v −→ w. ii. Show that the squares of the diagonals are |O R | 2 = | −→ v + −→ w | 2 = | −→ v | 2 + 2 −→ v · −→ w + | −→ w | 2 and | PQ | 2 = | −→ v −→ w | 2 = | −→ v | 2 2 −→ v · −→ w + |
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Unformatted text preview: −→ w | 2 . iii. Show that |O R | 2 + | PQ | 2 = 2 | −→ v | 2 + 2 | −→ w | 2 ; but of course |O P | 2 + | PR | 2 + | RQ | 2 + | Q O| 2 = | −→ v | 2 + | −→ w | 2 + | −→ v | 2 + | −→ w | 2 = 2 | −→ v | 2 + 2 | −→ w | 2 . 5. Show that if −→ v = x −→ ı + y −→ is any nonzero vector in the plane, then the vector −→ w = y −→ ı − x −→ is perpendicular to −→ v ....
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