Adding Vectors Mathematically Adding vectors graphically works, but it is slow, cumbersome, and slightly unreliable. Adding vectors mathematically allows us to ditch the protractor and bring the precision of mathematics to give us the same answer every time.

9 4 = = 2

59 30 = = 30° 29 30° °

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What about more than two vectors?

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What is similar about these problems? What makes them easy?

Vectors that are parallel or perpendicular are EASY to add together mathematically. If parallel – add/subtract their magnitudes. If perpendicular – move them to form a right triangle. Pythagorean Theorem  magnitude of resultant Inverse trig function  angle.

Which is greater: 6+5 or 11? Which is greater:.5 or ½ or 4/8? These are all the same thing, except written differently. We do the same thing with vectors.

° 30° F 1x F 1y F 2x F 2y F 1 = 40 F 2 =10 10° 30°

° 30° F 1x F 1y F 2x F 2y

40 10° 10 30° F 1x F 1y F 1 = 40 10°

We can decompose the vectors into x and y components (parts). These parts will always be parallel or perpendicular to the other parts. Then we can easily add them.

Draw the x- and y-components of each vector. F2F2 F4F4 F1F1 F3F3

WS 3 Only do a) through d) for now.

A trig shortcut

° 30° F 1x F 1y F 2x F 2y F 1 = 40 F 2 =10 10° 30°

θ F F y = F sin (θ) F x = F cos (θ)

θ F F y = -F sin (θ) F x = F cos (θ)

θ F F x = -F sin (θ) F y = F cos (θ)

θ F

θ F

θ F

θ F

Now finish WS 3 using the shortcuts.

Review It’s easy to mathematically add vectors if they are parallel or perpendicular. If they aren’t, we can break them down into x and y components (parts). Why is this advantageous? Doesn’t it just make two vectors in replace of one?