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

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?

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

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

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.

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

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

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

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

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

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

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 = R y = R = ??

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

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 = km  R = km

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

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

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

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

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 = R y = R = 32.33

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

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 = R y = R = θ

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?

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

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.

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.

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

Adding Vectors by the Component Method  Other examples A duck flies 10 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

Adding Vectors by the Component Method  Other examples A duck flies 10 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

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

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

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

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

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.