Engineering Calculus Notes 43

# Engineering Calculus Notes 43 - (b Now suppose −→ u...

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1.2. VECTORS AND THEIR ARITHMETIC 31 (c) Show that if a −→ v = −→ 0 then either a = 0 or −→ v = −→ 0 . ( Hint: What do you know about the relation between lengths for −→ v and a −→ v ?) (d) Show that if a vector −→ v satisfies a −→ v = b −→ v where a negationslash = b are two specific, distinct scalars, then −→ v = −→ 0 . (e) Show that vector subtraction is not associative. 4. (a) Show that if −→ v and −→ w are two linearly independent vectors in the plane, then every vector in the plane can be expressed as a linear combination of −→ v and −→ w . ( Hint: The independence assumption means they point along non-parallel lines. Given a point P in the plane, consider the parallelogram with the origin and P as opposite vertices, and with edges parallel to −→ v and −→ w . Use this to construct the linear combination.)
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Unformatted text preview: (b) Now suppose −→ u , −→ v and −→ w are three nonzero vectors in R 3 . If −→ v and −→ w are linearly independent, show that every vector lying in the plane that contains the two lines through the origin parallel to −→ v and −→ w can be expressed as a linear combination of −→ v and −→ w . Now show that if −→ u does not lie in this plane, then every vector in R 3 can be expressed as a linear combination of −→ u , −→ v and −→ w . The two statements above are summarized by saying that −→ v and −→ w ( resp . −→ u , −→ v and −→ w ) span R 2 ( resp . R 3 ). Challenge problem:...
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