You are watching: Decompose the vector bb into a component parallel to aa and a component orthogonal to aa.

**number 1 shows vectorns u**and also

**v**via vector

**u**dewritten right into orthogonatogether Components w1 and w2.

**Vector u**have the right to now it is in written u = w1 + w2, wbelow w1 ins paralletogether come vector

**v**and also w1 is perpendicular/orthogonatogether come w2. The vector ingredient w1 is likewise dubbed the forecast of vector

**u**oncome vector

**v**, projv

**u**.the projv u deserve to it is in calculation as follows:

forecast the U ONcome V:

**Lens u** and also **v** it is in nonzero vectors:

pro j v u=< u·v v 2 >v

as soon as ns vector component of projv uis found, Due to the fact that u = w1 + w2, component vector w2 have the right to it is in found by subtracting w1 native

**u**.

w2 = u - w1

Let"ns look at in ~ part examples.come work this Instances needs ns usage the various vector rules. If you are not familiar via a preeminence walk to the connected subject because that a review.

instance 1: Lens u=〈-2,2〉 and v=〈3,5〉. Create vector u as ns amount the two orthogonal vectors among i beg your pardon ins a forecast that u onto v.

agree j v u=< u·v ∥v ∥ 2 >v= w 1 | pro j v u=< u·v ∥v ∥ 2 >v pro j v u=< ( −2·3 )+( 2·5 ) 3 2 + 5 2 2 >〈3,5〉 agree j v u=< −6+10 34 2 >〈3,5〉 agree j v u=< 4 34 >〈3,5〉=< 2 17 >〈3,5〉 pro j v u=〈 6 17 , 10 17 〉 |

step 2: uncover the orthogonal component. w2 = u - w1 | w2 = u - w1 w 2 =〈−2,2〉−〈 6 17 , 10 17 〉 w 2 =〈( −2− 6 17 ), ( 2− 10 17 )〉 w 2 =〈− 40 17 , 24 17 〉 |

step 3: compose ns vector as ns sum that two orthogonatogether vectors. u = w1 + w2 | u = w1 + w2 u=〈 6 17 , 10 17 〉+〈− 40 17 , 24 17 〉 |

instance 2: offered vector u=〈1,3〉 and v=〈-4,5〉, compose u together a amount the two orthogonatogether vectors, one i beg your pardon is a estimate the u onto v.

agree j v u=< u·v ∥v ∥ 2 >v= w 1 | agree j v u=< u·v ∥v ∥ 2 >v agree j v u=< ( 1·−4 )+( 3·5 ) ( −4 ) 2 + 5 2 2 >〈−4,5〉 agree j v u=< −4+15 41 2 >〈−4,5〉 agree j v u=< 11 41 >〈−4,5〉 agree j v u=〈− 44 41 , 55 41 〉 |

action 2: discover the orthogonatogether component. w2 = u - w1 | w2 = u - w1 w 2 =〈1,3〉−〈− 44 41 , 55 41 〉 w 2 =〈( 1+ 44 41 ), ( 3− 55 41 )〉 w 2 =〈 85 41 , 68 41 〉 |

step 3: create the vector as ns sum the two orthogonatogether vectors. u = w1 + w2 | u = w1 + w2 u=〈− 44 41 , 55 41 〉+〈 85 41 , 68 41 〉 |

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