PHY183-Lecture04-1 - Vector Subtraction For every vector A...

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1 January 12, 2012 Physics for Scientists&Engineers 1 1 Physics for Scientists & Engineers 1 Spring Semester 2012 Lecture 4 – Vectors and Problem Solving January 12, 2012 Physics for Scientists&Engineers 1 2 For every vector there is a vector , with the same length, pointing in the exact opposite direction Vector subtraction: To obtain the vector , we add the vector to , following the procedure of vector addition. Vector Subtraction x y A B A A A A B D A = A B D A = A B ()0 AA +− =  January 12, 2012 Physics for Scientists&Engineers 1 3 In Vector Subtraction Order Matters Reverse the order and take instead of . What is the result? The resulting vector is exactly the opposite vector to Rules for vector addition and subtraction are just like for real numbers. x y B D A = A B x y A E B = A B B A A B A E B = B D A = January 12, 2012 Physics for Scientists&Engineers 1 4 Unit Vectors Vector representation in terms of unit vectors: 2D case ˆˆ xy Aa xa y =+ ˆ xyz ya z =++ x y A ˆ x ax a y ˆ y ˆ y ˆ x a y a x Projection of on the y axis provides its component A a y ˆ (1,0,0) ˆ (0,1,0) ˆ (0,0,1) x y z = = = January 12, 2012 Physics for Scientists&Engineers 1 5 Component Method for Vector Addition Vector addition can also be accomplished by using Cartesian components and unit vectors. Component representation Vector addition Components of sum vector with [] () ˆ ˆ ˆ ˆ ˆ ˆ ˆ xyz xyz xx yy zz CAB aaa bbb ab y x b z z ba z a =++++ + =+ ++ ++ c x = a x + b x c y = a y + b y c z = a z + b z ˆ ˆ ˆ Cc c z x y c ˆ ˆ ˆ ˆ ˆ ˆ z x x Aa a a Bb b b z y y January 12, 2012 Physics for Scientists&Engineers 1 6 Addition of Two 2D Vectors A = A x ˆ x + A y ˆ y B = B x ˆ x + B y ˆ y C = C x ˆ x + C y ˆ y = A + B = ( A x + B x ) ˆ x + ( A y + B y ) ˆ y
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2 January 12, 2012 Physics for Scientists&Engineers 1 7 Vector Subtraction Procedure is exactly the same as vector addition Difference vector: With components: [] () ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ xyz xyz xx yy zz DAB aaa bbb ab y x b z z ba z a =− =++−+ + =− +− +−  d x = a x b x d y = a y b y d z = a z b z ˆ ˆ ˆ xyz Dd d z x y d =++ ˆ ˆ ˆ ˆ ˆ ˆ z x x Aa a a Bb b b z y y with An equation between vectors equals three scalar equations! January 12, 2012 Physics for Scientists&Engineers 1 8 Multiplication of a Vector with a Scalar Let imagine adding the same vector to itself three times The resulting vector is three times longer and points in the same direction as the original vectors For multiplication of a vector with a scalar, we obtain The components are A + A + A E = s A = s ( A x , A y , A z ) = ( sA x , sA y , sA z ) E x = sA x E y = sA y E z = sA z January 12, 2012 Physics for Scientists&Engineers 1 9 Vector length and direction Vector in component representation (in 2D)
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This note was uploaded on 04/02/2012 for the course PHY 183 taught by Professor Wolf during the Spring '08 term at Michigan State University.

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PHY183-Lecture04-1 - Vector Subtraction For every vector A...

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