PHY183-Lecture3

# PHY183-Lecture3 - Physics for Scientists &amp;amp;...

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Unformatted text preview: Physics for Scientists & Engineers 1 Spring Semester 2006 Lecture 3 January 11, 2006 Physics for Scientists&Engineers 1 1 Vectors Vectors are heavily used in physics Need to manipulate them without any difficulty r Vector C Beginning and end point Characterized by: Magnitude Direction Unit Remark: a quantity defined without a direction is a scalar January 11, 2006 Physics for Scientists&Engineers 1 2 Cartesian Coordinate system (1) Used as a representation of vectors Quantifies a direction in 2-dimensional space Two perpendicular directions x to the right, y upward Position of point P specified by (P ,P ) x y Px and Py are positive or negative real numbers January 11, 2006 Physics for Scientists&Engineers 1 3 Cartesian Coordinate system (2) Quantifies a direction in 3-dimensional space Third direction coming straight out of page Higher dimensionalities are used in modern theories (but quite abstract and hard to represent on a twodimensional paper) January 11, 2006 Physics for Scientists&Engineers 1 4 Right-Hand Rule Conventional assignment for right-handed coordinate system (more on 3d coordinate systems later in this semester) January 11, 2006 Physics for Scientists&Engineers 1 5 Cartesian coordinate system (3) r A joins P and Q Q = (3,1) P = (-2,-3) r A = (3 - (-2),1 - (-3)) = (5,4) Shift it to the origin for a simple representation r A joins R and S S = (2,3) R = (-3,-1) r A = (2 - (-3),3 - (-1) = (5,4) January 11, 2006 Physics for Scientists&Engineers 1 6 r r r C = A+ B Vector Addition - Graphical We learned: you can drag vectors around in space without changing their value Length stays the same Direction stays the same In particular, you can drag vector B in such a way that its foot is at the tip of vector A Sum vector C then points from the foot of A to the tip of B You can also do it the other way around: r r r r r C = A+ B= B+ A January 11, 2006 Physics for Scientists&Engineers 1 7 Vector Subtraction r r For every vector A there is a vector - A , with the same length, pointing in the exact opposite r r r r r direction -A A Vector subtraction: A + -A = 0 y ( ) r r r To obtain the vector rD = B - A , r we add the vector - A to B , following the procedure of vector addition. r A r B r -A x r r r D= B- A 8 January 11, 2006 Physics for Scientists&Engineers 1 Reverse the order and r r taker A - B instead of r B - A . What is the result? In Vector Subtraction Order Matters y r A r B r -A x r r r vector The resulting r r r D= B- A E = A-B y r r is exactly the opposite A r r r r -B B vector to D = B - A . Rules for vector addition and subtraction are just r r r x E = A-B like for real numbers. January 11, 2006 Physics for Scientists&Engineers 1 9 Unit Vectors Vector representation in terms of unit vectors: r ^ ^ ^ A = ax x + ay y + az z ^ x = (1,0,0) ^ y = (0,1,0) ^ z = (0,0,1) 2d case Projection of ay on the y axis provides its component ay r A y ^ ay y ^ y r A ^ ax x ^ x r ^ ^ A = ax x + ay y January 11, 2006 x ax 10 Physics for Scientists&Engineers 1 Component Method for Vector Addition Vector addition can also be accomplished by using Cartesian components and unit vectors. r Component representation ^ ^ ^ A = ax x + ay y + az z r ^ ^ ^ B = bx x + by y + bz z Vector addition r r r C = A+ B r ^ ^ ^ ^ ^ ^ C = [ax x + ay y + az z ] + [bx x + by y + bz z ] r ^ ^ ^ C = (ax + bx ) x + (ay + by ) y + (az + bz ) z Components of sum vector r ^ ^ ^ C = cx x + cy y + czz January 11, 2006 with c x = ax + bx c y = ay + by c z = az + bz 11 Physics for Scientists&Engineers 1 Vector Subtraction Procedure is exactly the same as vector addition: r ^ ^ ^ A = ax x + ay y + az z r ^ ^ ^ B = bx x + by y + bz z Difference vector: Components r r r D= A-B r ^ ^ ^ ^ ^ ^ D = [ax x + ay y + az z ] - [bx x + by y + bz z ] r ^ ^ ^ D = (ax - bx ) x + (ay - by ) y + (az - bz ) z r ^+ ^ ^ D = dx x dy y + dz z with dx = ax - bx dy = ay - by dz = az - bz An equation between vectors equals three scalar equations! January 11, 2006 Physics for Scientists&Engineers 1 12 Vector length and direction r Vector A in component representation (in 2D) r ^ ^ A = ax x + ay y Calculation of its norm (=length) from its components y ^ ay y ^ y r A q P Using Pythagoras in the right triangle OPQ Q r 2 2 A = ax + ay x ^ O x ^ ax x r Also, the angle q between A and the x axis q = arctan( ay /ax ) Physics for Scientists&Engineers 1 January 11, 2006 13 ...
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