PHY183-Lecture3 - Physics for Scientists &...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

Ask a homework question - tutors are online