Engineering Calculus Notes 77

# Engineering Calculus Notes 77 - point P in P can be...

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1.5. PLANES 65 are two linearly independent vectors in R 3 . If we represent them via arrows in standard position, they determine a plane P 0 through the origin. Note that any linear combination of −→ v and −→ w −→ p ( s,t ) = s −→ v + t −→ w is the position vector of some point in this plane: when s and t are both positive, we draw the parallelogram with one vertex at the origin, one pair of sides parallel to −→ v , of length s | −→ v | , and the other pair of sides parallel to −→ w , with length t | −→ w | (Figure 1.35 ). Then −→ p ( s,t ) is the vertex opposite −→ v −→ w t −→ w s −→ v −→ p ( s,t ) Figure 1.35: Linear Combination the origin in this parallelogram. Conversely, the position vector of any
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Unformatted text preview: point P in P can be expressed uniquely as a linear combination of −→ v and −→ w . We leave it to you to complete the details (see Exercise 4 in § 1.2 ). Remark 1.5.3. If −→ v and −→ w are linearly independent vectors in R 3 , then the set of all linear combinations of −→ v and −→ w P ( −→ v , −→ w ) := { s −→ v + t −→ w | s,t ∈ R } is the set of position vectors for points in the plane (through the origin) determined by −→ v and −→ w , called the span of −→ v and −→ w ....
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## This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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