Vector Notes video

Vector Notes: 

Vector Notes

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NO!

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A = B = sqrt (2)/2 * C = sqrt (2)/2 Break vector into components (red and blue lines)

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Vectors so far 1st Leg : < 0, 2 > 2nd Leg : < -0.707 , 0.707 How far away and in what direction from Where he started? Want to add the vectors together But <0 , 2> + < -0.707, 0.707 > = ?

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Vectors so far 1st Leg : < 0, 2 > 2nd Leg : < -0.707 , 0.707 > Add them up: Resultant : < -0.707 , 2.707 > ~0.7 miles West and 2.7 miles North Find Magnitude: Mag = ( (-0.707)^2 + (2.707)^2 )^.5 Mag = 3.05 miles

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Leg 1 < 0 , 2 > Leg 2 < -0.707, 0.707 > Leg 3 < 2, 0 > Leg 4 < 0.707, -0.707 Summing these up our Hiker is: 2 miles east and 2 miles north. Applying Pythagorean Theorem: Mag = ( (2)^2 + (2)^2 )^.5 Mag = 2.83 miles from his car Note: The order of vectors doesn’t matter! Doing the journey in a different order Takes you to the same place!

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Finding Direction of a Vector In general, direction is what angle the vector points from +X axis N, NW, W, SW, S, SE, E, and NE are just specific angles! Make Vector into triangle! SOH – Sine = Opposite / Hypotenuse OR: sin(theta) = Y / H CAH – Cosine = Adjacent / Hypotenuse OR: cos (theta) = X / H TOA – Tangent = Opposite / Adjacent OR: tan(theta) = Y / X

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SOH – Sine = Opposite / Hypotenuse OR: sin(theta) = Y / H CAH – Cosine = Adjacent / Hypotenuse OR: cos (theta) = X / H TOA – Tangent = Opposite / Adjacent OR: tan(theta) = Y / X Tan(theta) = Opposite / Adjacent = Y / X = 3 / 2 Arctan (tan(theta)) = arctan (3/2) Theta = arctan (3/2) = (from calculator) 56.3 degrees NOTE: check whether Your calculator is giving Degrees or radians! Finding Direction of a Vector

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Leg 1 < 0 , 2 > Leg 2 < -0.707, 0.707 > Leg 3 < 2, 0 > Leg 4 < 0.707, -0.707 > Summing these up our Hiker is: 2 miles east and 2 miles north. Applying Pythagorean Theorem: Mag = ( (2)^2 + (2)^2 )^.5 Mag = 2.83 miles from his car Theta = arctan (2/2) = 45 degrees from +X Theta = arccos (2/2.83) = 45 degrees ‘’ ‘’ ‘’ Theta = arcsin (2/2.83) = 45 degrees ‘’ ‘’ ‘’ Probably could have noticed this without a calculator!

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Recap Introduced Vectors Magnitude^2 = X_component ^2 + Y_component^2 Pythagorean Theorem – Right Triangle comparison Trig Identities for finding direction (SOH CAH TOA) SOH – Sine = Opposite / Hypotenuse OR: sin(theta) = Y / H CAH – Cosine = Adjacent / Hypotenuse OR: cos (theta) = X / H TOA – Tangent = Opposite / Adjacent OR: tan(theta) = Y / X

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Vectors Recap 2 Vectors add component-wise <1,2> + <3,5> = <1+3,2+5> = <4,7> Break vector into components: In the left hand-side case: X = H * cos (theta) Y = H * sin(theta) Careful: We will not always have the angle exactly this way!

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Vector from two points Hiker is 2 miles north and 2 miles east of Parking lot  < 2 , 2 > Helicopter is 3 miles south and 1 mile east of Parking lot  < 1 , -3 > Essentially know locations of two places. Need to find vector connecting the two. End Point – Start Point  (2,2) – (1,-3) < 1 , 5 > Magnitude = Distance = ( 1^2 + 5^2) ^.5 = (26)^.5 Direction: theta = arctan (5/1) = 78.69 degrees N of E Run through the problem on your Own as practice!