Phys 170 Lecture 5 1 Physics 170 Lecture 5 Dot Product: Projection of Vector onto a Line & Angle etween Two Vectors
Phys 170 Lecture 5 2 Mastering Engineering Introduction to M.E. and Assignment 1 moved to next Friday Assignment 2, that will appear at 6 PM tonight also due then. 196 / 236 students show up in the system Has anyone gotten the email with the actual access code? I made optional (zero points) assignments for Chapter 3: the Fundamental Problems with solutions in the book, and the Tutorials The due dates of the optional assignments don t matter.
Phys 170 Lecture 5 3 Calculator Linear Equation Instructions Linear Equations on calculators Casio fx-991ms This calculator can solve linear equations for 2 or 3 variables. Enter "equation mode" by pressing MODE MODE MODE then 1 (EQN). A small EQN at the top of the display indicates that you are in "equation mode" Going into "calc mode" or "vector mode" or "matrix mode" erases any equation information. Normal calculations are not possible in "equation mode," except to calculate equation coefficients. After entering "equation mode", the screen shows Unknowns? -> (right-cursor for quadratic or cubic equations) 2 3 Enter 3 for 3 unknowns, and the screen shows a1? 0. Enter the first coefficient for the first equation (or a calculation for it) and =(equals), and the screen shows b1? 0. Enter the second coefficient for the first equation (or a calculation for it) and =(equals), and the screen shows c1? 0. Enter the third coefficient for the first equation (or a calculation for it) and =(equals), and the screen shows d1? 0. Enter the right-hand-side constant for the first equation (or a calculation for it) and =(equals), and the screen shows a2? 0. Continue this process of entering coefficients and right-hand-sides for the second equation, and third equation. After the last value (right hand side of third equation) and =(equals), the screen shows x= xx.xxxxxxxxxxx the solution for the first variable. Hit =(equals) and the screen shows y= yy.yyyyyyyyyyy the solution for the second variable. Hit =(equals) and the screen shows z= zz.zzzzzzzzzzz the solution for the third variable. Hit =(equals) and the screen shows a1= aa.aa1 the first coefficient that you entered You can use the up and down cursors to scroll through the coefficients and right-hand-sides, to check or edit them. You can also use the cursors to scroll through the solutions. The AC (all-clear) button does not clear the coefficients, nor does turning the calculator off. Exiting "equation mode" and re-entering does clear them (sets them all to zero).
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Phys 170 Lecture 5 5 Components of F 1 20 downward from xy plane u 1Z = sin20 Projection of unit vector onto xy plane u 1XY = cos20 Rotated 25 from y axis u 1X = cos20 sin25 u 1Y = cos20 cos25 Unit vector is û 1 = cos20 sin25 î + cos20 cos25 ĵ sin20 ˆk = 0.3971î + 0.8517 ĵ 0.3420 ˆk I won t multiply by magnitude 3.21 kn yet...
Phys 170 Lecture 5 6 Components of F 2 65 away from z axis (γ angle) 50 away from y axis (β angle) u 2Z = cos65 = 0.4226 u 2Y = cos50 = 0.6428?? away from x axis (α angle) Use fact that u X 2 + u Y 2 + u Z 2 = 1 u X = ± 1 u Y 2 u Z 2 Get the sign from guide-lines in the figure: u2x < 0 = 0.6389 I don t need to multiply by the magnitude, because...
Phys 170 Lecture 5 7 Projection of One Vector on Another Vector Draw vectors A and tail to tail. A cosθ θ They form an angle θ. From the head of, draw the perpendicular cosθ A line to. The length A cosθ is the projection of A on. From the head of, draw the perpendicular line to. The length cosθ is the projection of on A A. A
Vector Dot Product Draw the two vectors tail to tail. They form an angle θ. θ θ A A The dot product is defined as The projection of A onto the direction of is A cosθ, and the projection of onto the direction of A is cosθ. So Ai is the magnitude of one vector times the projection of the other vector. d = Ai = A cosθ = A cosθ A cosθ Phys 170 Lecture 5 8 θ cosθ A
Vector Dot Product Draw the two vectors tail to tail. They form an angle θ. θ θ A A The dot product is defined as The projection of A onto the direction of is A cosθ, and the projection of onto the direction of A is cosθ. So Ai is the magnitude of one vector times the projection of the other vector. d = Ai = A cosθ = A cosθ A cosθ Phys 170 Lecture 5 9 θ cosθ A
Vector Dot Product Draw the two vectors tail to tail. They form an angle θ. θ θ A A The dot product is defined as The projection of A onto the direction of is A cosθ, and the projection of onto the direction of A is cosθ. So Ai is the magnitude of one vector times the projection of the other vector. d = Ai = A cosθ = A cosθ A cosθ Phys 170 Lecture 5 10 θ cosθ A
Phys 170 Lecture 5 11 Properties of Dot Product d = Ai = A cosθ = A cosθ The order doesn t matter: Ai = i A. If we doubled the length of A, we would double the product. If we doubled the length of, we would double the product. ( ca )i = Ai ( c ) = c( Ai ) If θ < 90, cosθ > 0, so Ai > 0 If θ = 90, cosθ = 0, so Ai = 0! If θ > 90, cosθ < 0, so Ai < 0!!
Dot Product from Components If we know the components of two vectors, there s a very nice formula for computing the dot product. It s just Ai = A x x + A y y + A z z In two dimensions, leave off the z-components (call them zero). It s not obvious that this is equivalent to the geometric definition of the dot-product. ut it is. If you write A and in ijk form, and use distributive law and the fact that the Cartesian unit vectors are perpendicular so their dot product is zero, it s not hard to prove. Phys 170 Lecture 5 12
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Phys 170 Lecture 5 14 Projection of F 1 on Line of F 2 F 1 V = (projection of F 1 on V ) V If we use the unit vector of F2, its magnitude is 1, so we don t need to divide. That s why I didn t multiply by the magnitude of F2. It s enough to do the dot-product of the unit-vectors, then multiply that by the magnitude of F1. (+0.3971, +0.8517, 0.3420) ( 0.6389, +0.6428, +0.4226) 0.2537 + 0.5475 0.1445 = 0.1492 0.1492 3.21 kn = 479 N
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Phys 170 Lecture 5 16 Angle etween Vectors If we are given two vectors, especially in 3D, doing the geometry to figure out the angle between the vectors is hard. ut it s easy using the dot-product. Find the components of both vectors, and do the dot-product Ai = A Since Ai = A x x + A y y + A z z cosθ, if we divide the dot-product by A what s left is cosθ, and we just take the inverse-cosine. So θ = cos 1 Ai A A = cos 1 x x + A y y + A z z A 2 x + A 2 2 y + A z 2 x + 2 2 y + z
Phys 170 Lecture 5 17 Angle etween F 1 and F 2 F 1 F 2 = F 1 F 2 cosθ ut since we have the unit vectors, û 1 û 2 = 1 1 cosθ! And we already know the dot product: 0.1492. cos 1 ( 0.1492) = 81.4
Phys 170 Lecture 5 18 For Next Time Read all of Chapter 3 Free body diagrams Point equilibrium in 2D and 3D and Chapter 4 sections 1-3 Moments in 2D (several ways to do it) Cross product of vectors Moments in 3D (almost always using cross product) Homework assignment not due until next Friday. Practice problems & tutorials available (optional, no points)