6.05 - Linear Combinations - Edubirdie

University:
High School
Course:
High School Physics
Academic year:

2024
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6
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Linear Combinations ◼ Since a vector can be written as the sum of vectors (each perhaps multiplied by a scalar), a vector can be expressed as a linear combination of other vectors. w = k1 u + k 2 v ◼ Ex. w = 2u + 3v ◼ To solve for scalars, use a sys of equations Geometric Approach ◼ ◼ Ex. Express vector w = (16, 26 ) as a linear combination of vectors u = (2, 5) & v = (3, 4) w = k1 u + k 2 v (16, 26 ) = k1 (2 , 5) + k 2 (3 , 4) 16 = 2k1 + 3k 2 x5 26 = 5k1 + 4k 2 x -2 80 = 10k1 + 15k 2 + − 52 = − 10k1 + − 8k 2 28 = 7k 2 k2 = 4 k1 = 2 w = 2u + 4v ◼ s = 32cm, 45o t = 9 cm,112o r = 208 cm, 90o Ex. Express vector r as a linear combination of vectors s & t Need components! r = k1 s + k 2 t = 22.63k1 + − 3.37k 2 0 208 = 22.63k1 + 8.34k 2 0 x -1 = 22.63k1 + − 3.37k2 − 208 = − 22.63k1 + − 8.34k2 + − 208 = − 11 .71k 2 s = (22 .63 , 22 .63) t = (−3.37 , 8.34) r = (0 , 208 ) k 2 = 17 .76 k1 = 2.64 r = 2.64 s + 17 .76t

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6.05 - Linear Combinations

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