6.01 - Intro to Vectors

University:
High School
Course:
High School Physics
Academic year:

2024
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13
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6.01 Intro to Vectors ◼ A scalar quantity is a quantity that is entirely defined by a number (with unit) Ex. mass of 4kg ◼ Ex. cost of $3.62 ◼ ◼ A vector quantity is a quantity that is defined by a number & a direction Ex. travel a distance of 200km NE ◼ Ex. wind blows 80km/h from the south ◼ ◼ A vector represents both magnitude & direction  ◼ Written: u or  v or  u AB or etc… AB etc… ◼ Magnitude or norm: ◼ Represented geometrically or algebraically ◼ Geometric Description:  vector shown as arrow, length represents the norm & the direction is relative to positive xaxis.  Describe with magnitude and direction. B arrowhead (tip)  Ex Vector: AB A tail 5 cm AB = 5 cm 45o Direction: 45o tail 250o  Ex arrowhead (tip) v 2 km Vector: v v = 2 km Direction: 250o ◼ Algebraically:  vector represented as an ordered pair (a, b)  tail has coordinates (x1, y1)  arrowhead has coordinates (x2, y2)  (a, b) represents the horizontal component a = x2 – x1 & the vertical component b = y2 – y1 of the vector  Ex a = x2 – x1 b = y2 – y1 Vector: u = (5, − 3) (-3, 2) u (2, -1) a = (2 - -3) b = (-1 - 2) a=5 b = -3 ◼ To find a vector’s norm, direction or components: ◼ Measure (if to scale) (geometric method) ◼ Pythagoras using components (algebraic method) ◼ Distance formula using coordinates of tail/head (algebraic method) d = ( x − x )2 + y − y 2 2 ◼ ( 2 Trig (algebraic or geometric method)  used when adding vectors 1 SOH CAH TOA (often with components) a b c = =  Sine Law: sin A sin B sin C  Cosine Law: c 2 = a 2 + b2 − 2ab cos C 1 ) a b Given vector w = (3, 6 ) find the norm & direction w = 3 +6 2 w 6  3 w = 45 2 6 tan = 3  = 63 o Given vector v with a tail at (1, 8) and an arrowhead at (6, 2) find the norm & direction a=x –x b=y –y 2 (1, 8)  v 5 1 2 a = (6 - 1) b = (2 - 8) a=5 b = -6 -6 v = 5 + ( −6) 2 (6, 2) 1 v = 61 OR distance formula!!! 2 v = (5, − 6) 6 tan = 5  = 50 o  = 310 o AB = 6 at a direction of 220°. Solve for the components. 220o a a cos 40 = 6 b sin 40 = 6 a = – 4.60 b = – 3.86 o 40 o b 6 AB o AB = (− 4.60, − 3.86 ) A unit vector has a direction of 240°. Solve for the components. 240o 60 o a cos 60 = 1 b sin 60 = 1 1 a = – 0.5 b = – 0.866 OR − 3 a o b o  3  u =  − 0.5, −  2   Unit Vector has a magnitude of 1 2 Try It ◼ Try It ◼ Try It ◼