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Adding Vectors by the Component Method Feel free to use to accompanying notes sheet.

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Adding Vectors by the Component Method Yesterday we added vectors which were at right angles to one another. What would happen if the vectors were not at right angles? A crow (who apparently isn’t aware that he should only fly in straight lines) first flies 10 km at 60° N of E. The he flies 25 km at 10° N of E. What is his total displacement?

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Adding Vectors by the Component Method – The Strategy A = 10 km @ 60° N of E B = 25 km @ 10° N of E Obviously the answer is not 35 km, so what must be done? 60° 10°

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Adding Vectors by the Component Method A = 10 km @ 60° N of E B = 25 km @ 10° N of E Obviously the answer is not 35 km, so what must be done? Find the RESULTANT R = resultant

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Adding Vectors by the Component Method The vectors need to be added vectorally. Both vectors need to be RESOLVED into their components. The components are then added together to find the resultant. A to be resolved into A x and A y. B to be resolved into B x and B y.

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Adding Vectors by the Component Method – Resolve each vector into components A = 10 km @ 60° N of E B = 25 km @ 10° N of E

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Adding Vectors by the Component Method – Resolve each vector into components A = 10 km @ 60° N of E B = 25 km @ 10° N of E AyAy AxAx ByBy BxBx

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Adding Vectors by the Component Method A x = 10cos60° A y = 10sin60° B x = 25cos10° B y = 25sin10° A = 10 km @ 60° N of E B = 25 km @ 10° N of E AyAy AxAx ByBy BxBx

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Adding Vectors by the Component Method The x-components and y-components can each be considered legs of the resulting triangle. AyAy AxAx ByBy BxBx

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Adding Vectors by the Component Method The x-components and y-components can each be considered legs of the resulting triangle. AyAy AxAx ByBy BxBx

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Adding Vectors by the Component Method – Construct the “resulting triangle” from components R x = A x + B x R y = A y + B y AyAy AxAx ByBy BxBx R RxRx RyRy

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Adding Vectors by the Component Method How do we find R? YES!! Pythagorean Theorem: R 2 = R x 2 + R y 2 AyAy AxAx ByBy BxBx R RxRx RyRy R x = 29.60 R y = 13.00 R = ??

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Adding Vectors by the Component Method R 2 = R X 2 + R Y 2 AyAy AxAx ByBy BxBx R RxRx RyRy

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Adding Vectors by the Component Method Math – IN DEGREE MODE A x = 10cos60° = 5.00 km A y = 10sin60° = 8.66 km B x = 25cos10° = 24.6 km B y = 25sin10° = 4.34 km R x = A x + B x = 29.6 km R y = A y + B y = 13.00 km R = 32.33 km

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Adding Vectors by the Component Method Another way to organize data x-axisy-axis A 10cos60°10sin60° B 25cos10°25sin10° R

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Adding Vectors by the Component Method Another way to organize data x-axisy-axis A 5.008.66 B 24.624.34 R

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Adding Vectors by the Component Method Another way to organize data x-axisy-axis A 5.008.66 B 24.624.34 R 29.62 13.00

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Adding Vectors by the Component Method Is this it? 32.33 km? AyAy AxAx ByBy BxBx R RxRx RyRy R x = 29.60 R y = 13.00 R = 32.33

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Adding Vectors by the Component Method Is this it? No!! Now we have to find the direction. AyAy AxAx ByBy BxBx R RxRx RyRy R x = 29.60 R y = 13.00 R = 32.33

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Adding Vectors by the Component Method tan θ = R y /R x θ = tan -1 (R y /R x) R RxRx RyRy R x = 29.60 R y = 13.00 R = 32.33 θ

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Adding Vectors by the Component Method θ = 23.71° N of E How do we know it is North of East? R RxRx RyRy R x = 29.60 R y = 13.00 R = 32.33 θ

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Adding Vectors by the Component Method E (east) N (north) S (South) W (west) 40 ° N of E A B C Which angle is 10° North of West? A, B or C? Which angle is 10° West of North? A, B or C? What is angle C?

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Adding Vectors by the Component Method A = 10 km @ 60° N of E B = 25 km @ 10° N of E Final answer: 32.33 km @23.71° N of E R

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Adding Vectors by the Component Method Some helpful hints! Never use the original values after the vector has been resolved. Assign negative values to S and W components of vectors (assuming N and E are positive) Always make sure you are in degree mode.

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Adding Vectors by the Component Method Some helpful hints! Always make sure you are in degree mode. Make sure you draw your vectors in the correct directions initially. To help, redraw a coordinate system at the end of each vector. Be organized, stay organized, & finish the entire problem.

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Adding Vectors by the Component Method Other examples A duck flies 10 m/s @ 30° S of W with a wind blowing 5 m/s N. What is the resulting velocity of the duck? Answer:

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Adding Vectors by the Component Method Other examples A duck flies 10 m/s @ 30° S of W with a wind blowing 5 m/s N. What is the resulting velocity of the duck? Answer: A -x & -y B 0 x & +y

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Adding Vectors by the Component Method Other examples A duck flies 10 m/s @ 30° S of W with a wind blowing 5 m/s N. Answer: R = ?? x-axisy-axis A -10cos30-10sin30 B 0+5 R A B

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Adding Vectors by the Component Method Other examples A duck flies 10 m/s @ 30° S of W with a wind blowing 5 m/s N. Answer: R = ?? x-axisy-axis A -8.66-5 B 0+5 R A B

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Adding Vectors by the Component Method Other examples A duck flies 10 m/s @ 30° S of W with a wind blowing 5 m/s N. Answer: R = ?? x-axisy-axis A -8.66-5 B 0+5 R -8.66 0 A B

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Adding Vectors by the Component Method Other examples A duck flies 10 m/s @ 30° S of W with a wind blowing 5 m/s N. Answer: R = -8.66 m/s or 8.66 m/s W x-axisy-axis R -8.66 0 A B

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Adding Vectors by the Component Method ANY and ALL vectors can (and will) be analyzed this way. Displacements Velocities Accelerations Forces Momentum

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Adding Vectors by the Component Method Any questions? These notes will be online You MUST be good at vectors to succeed / pass this class. Ask questions (in class) whenever necessary.

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