6.06 - Scalar Product

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

2024
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158

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18
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Scalar Product ◼ The scalar product of two vectors is a scalar quantity that corresponds to the sum of the products of their components If… u = (a, b ) v = (c, d ) then… u • v = ac + bd use if have components of each vector ◼ Ex Given vectors, calculate the scalar product v = (− 1, 6) a) u = (− 5, 2 ) u • v = ac + bd u • v = (−5)(−1) + (2)(6) u • v = 17 ( b) u = 3 2 , 4 ) v= ( 3, 6 ) u • v = ac + bd u • v = (3 2 )( 3 ) + (4)( 6 ) u•v=7 6 ◼ The scalar product is also the product of 1) the norm of a vector with 2) the norm of the orthogonal projection of another vector onto it. B A C  u • v = u  v cos P Scalar product = AP  AC use if have magnitude of each vector & angle between the vectors if placed tail-tail Work is the scalar product of force & displacement ◼ Ex ◼ A force of 2000N at 30° is applied to an object to displace it 500m. How much work was done? F = 2000 N 30 o d = 500 m F • d = F  d cos F • d = (2000 )(500 ) cos 30 o F • d = 866 ,025 J In physics… Components are orthogonal projection onto displacement vector FH cos 30 = 2000 o FH = 2000 cos 30 o H F = 2000 N FH • d = (2000 ) cos 30 (500 ) o 30 o A F • d = 866 , 025 J ◼ Ex 3 2 Calculate the scalar product a) u • v = u  v cos u • v = (2)(3) cos 0o u•v=6 6 b) 70 o 10 u • v = u  v cos u • v = (10)(6) cos 70 u • v = 20 .52 o ◼ Ex Given vectors, calc the scalar product &  (3, 5) a) u • v = ac + bd u = (2, 4 ) (1, 1) (3, 0) u • v = u  v cos 0 = ( 20 )( 5 ) cos   = 90 o v = (2, − 1) u • v = (2)(2) + (4)(−1) u•v=0 Any two orthogonal vectors have a zero scalar product! (3, 4) b) u • v = ac + bd (6, 2) (1, 1) u = (2, 3) v = (5,1) u • v = (2)(5) + (3)(1) u • v = u  v cos 13 = ( 13 )( 26 ) cos   = 45 o u • v = 13 ◼ Ex The scalar product of d and m is 138. If their magnitudes are 7 & 25 units, what is the value of  between the vectors? u • v = u  v cos 138 = (7)(25) cos   = 38 o ◼ Ex Find the acute  formed by lines: y = −2 x + 3 y = 2x +1 Find components of vector projected on each line u = (1, 2 )  v = (1, − 2) u • v = ac + bd = u  v cos (1)(1) + (2)(−2) =  = 126 .87 o So…. 5  5 cos  = 53 .13 o Multiple possible vectors since all pts on the line will yield same components since slope is constant! Try… P48 ◼ #3, #5b,c,e,h, #7a,c, #8 ◼ Scalar Product cont… ◼ Properties: ◼ ◼ ◼ u •v = v•u ( ) k u • k v = k k (u • v ) ku • v = k u • v 1 2 1 2 ◼ Ex a) Simplify 10u • 2v b) − 3u • 2u ◼ Ex If a = (− 1, 2 ) & b = (− 3, − 5) find the components of… a) (4a • 2b)(− 1a ) b) (a • b)(a + b)+ 3b − 2a ◼ Ex Consider u = (a1 ,b1 ) , v = (a2 ,b2 ) & u − v by the arrows OA , OB and BA A (a1, b1) u u −v Prove: u • v = a1a2 + b1b2 B (a2, b2) 2 2 2 u − v = u + v − 2 u  v cos 2 2 2 u  v cos = u + v − u − v 2 2 2 1 2  u  v cos =  u + v − u − v  2  2 2 1 2  u•v =  u + v − u −v  2  1 2 2 2 2 2 2 u • v = a1 + b1 + a2 + b2 − (a1 − a2 ) + (b1 − b2 ) 2 (( (( ) ( ) ( ) ( ) (( ) ( 1 2 2 2 2 2 2 2 2 u • v = a1 + b1 + a2 + b2 − a1 − 2a1a2 + a2 + b1 − 2b1b2 + b2 2 1 u • v = (2a1a2 + 2b1b2 ) 2 u • v = a1a2 + b1b2 )) )))