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Topic 1.3 Extended A - Vector Addition and Subtraction

Topic 1.3 Extended A - Vector Addition and Subtraction. Topic 1.3 Extended A - Vector Addition and Subtraction.  Unfortunately, objects move. If they didn't, finding the big three would be a cinch.  As it is, most of the things we interact with, in fact, we ourselves, move.

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Topic 1.3 Extended A - Vector Addition and Subtraction

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  1. Topic 1.3 ExtendedA - Vector Addition and Subtraction

  2. Topic 1.3 ExtendedA - Vector Addition and Subtraction Unfortunately, objects move. If they didn't, finding the big three would be a cinch. As it is, most of the things we interact with, in fact, we ourselves, move. And, to make matters worse, we move in more than one dimension. In this section we will look at vectors. A vector is a quantity that has both magnitude (size) and direction.

  3. Topic 1.3 ExtendedA - Vector Addition and Subtraction The direction of particle moving along a line is given by either a + sign (moving in the positive direction) or a - sign (moving in the negative direction). Thus, if a particle is traveling at 2 ft/s along the x-axis, it is moving in the positive x-direction. If a particle is traveling at -8 ft/s along the y-axis, it is moving in the negative y-direction. However, if a particle is moving in the x-y plane, NOT ALONG EITHER AXIS, its direction cannot be given with a simple sign. x y Instead, we need an arrow to show its direction. Such an arrow is called a vector. y A vector has both magnitude (numerical size), and direction. y vector Not all physical quantities have a direction. For example, pressure, time, energy, and mass do not have direction. Non-directional quantities are called scalars. x x

  4. FYI: The displacement from Sheboygan to Milwaukee has the same magnitude. What is different about it? Topic 1.3 ExtendedA - Vector Addition and Subtraction Note that the displacement vector traces out the SHORTEST distance between two points. The simplest vector is called the displacement vector. The displacement vector is obtained by drawing an arrow from a starting point to an ending point. For example, starting at Milwaukee and ending in Sheboygan would have a displacement vector that looks like this: actual path Note that the displacement vector (black) does not necessarily show the actual path followed (purple). displacement vector A displacement is simply a directed change in position, with disregard to the route taken.

  5. √ √ √ √ √ √ Topic 1.3 ExtendedA - Vector Addition and Subtraction The magnitude (size) of a displacement on the x-y plane is easy to obtain using the Pythagorean theorem or the distance formula: d = (x2 - x1)2 + (y2 - y1)2distance formula For example, the displacement shown has a magnitude given by d = (x2 - x1)2 + (y2 - y1)2 = (5 - 1)2 + (4 - 1)2 = 42 + 32 = 16 + 9 = 25 = 5 ft. y (ft) end (x2 , y2) = (5 , 4) (x1 , y1) = (1 , 1) x (ft) start

  6. Topic 1.3 ExtendedA - Vector Addition and Subtraction Of course, a scale drawing would allow you to measure the magnitude of the displacement directly with a ruler: Note that the length of the arrow is 5 ft. y (ft) end (x2 , y2) = (5 , 4) (x1 , y1) = (1 , 1) x (ft) start

  7. Topic 1.3 ExtendedA - Vector Addition and Subtraction Consider two vectors: vector a and vector b. In print, vectors are designated in bold non-italicized print. For our purposes, when we are writing a vector symbol on paper, use the letter with an arrow symbol over the top, like this: Each vector has a tail, and a tip (the arrow end). b tip tail a tip tail

  8. Topic 1.3 ExtendedA - Vector Addition and Subtraction Suppose we want to find the sum of the two vectors a + b = s. We take the first-named vector a, and translate the second-named vector b towards the first vector, SO THAT THE TAIL OF b CONNECTS TO THE TIP OF a. We are giving the sum an arbitrary name - say s. The result of the sum, which we are calling the vector s, is gotten by drawing an arrow from the tail of a to the tip of b. "translate" means to move without rotation. b tip tail a tip tip s tail tail

  9. Topic 1.3 ExtendedA - Vector Addition and Subtraction We can think of the sum a + b = s as the directions on a pirate map: Arrgh, matey. First, pace off the first vector a. Then, pace off the second vector b. And ye'll be findin' a treasure, aye!

  10. Topic 1.3 ExtendedA - Vector Addition and Subtraction We can think of the sum a + b = s as the directions on a pirate map: We start by pacing off the vector a, and then we end by pacing off the vector b. The treasure is at the ending point. The vector s represents the shortest distance to the treasure. s a b b = + end a s start

  11. Topic 1.3 ExtendedA - Vector Addition and Subtraction Now, suppose we want to subtract vectors. Just as you learned how to subtract integers by "adding the opposite," so, too, will we subtract vectors. Thus the difference of two vectors a - b is given by a - b = a + -b. difference of two vectors We just have to define the "opposite of b" or "-b." The opposite of a vector is the vector that is the same length, but points in the opposite direction. -b b the vector b a a+-b Thus, the opposite of the vector b -b a a-b = + -b

  12. Topic 1.3 ExtendedA - Vector Addition and Subtraction Observe the addition of vectors is commutative: a + b = b + a Observe that addition of vectors is associative: (a + b) + c = a + (b + c) b b a (a+b) c a a+b (a+b)+c b a c b+a (b+c) a a+(b+c) b

  13. N (pc) E (pc) Topic 1.3 ExtendedA - Vector Addition and Subtraction A pirate takes 6 paces east, 4 paces north, 2 paces west, and 1 pace south. (a) Draw a vector diagram showing the pirate's path. 13 paces (b) How far does the pirate walk? (c) Draw in the displacement vector. 4 paces 2 paces 1 pace 13 paces 6 paces + + + = 5 paces (d) What is the magnitude of the displacement? 2 paces 1 pace 4 paces 6 paces

  14. hypotenuse opposite θ adjacent trigonometric ratios Topic 1.3 ExtendedA - Vector Addition and Subtraction Suppose you know a vector's magnitude and direction. d dy = dsinθ θ You can find its components using the trigonometric functions: dx = dcosθ dy dx dy opp hyp adj hyp opp adj sinθ = cosθ = tanθ = d d dx d dy = dsinθ s-o-h-c-a-h-t-o-a dx = dcosθ

  15. Topic 1.3 ExtendedA - Vector Addition and Subtraction y(+) Recall the four quadrants of the Cartesian coordinate system: I II Quadrant xy x(-) x(+) I + + III IV II - + y(-) III - - IV + - Components will "inherit" the signs of the quadrants. For example, a vector in Quad II will have a negative x-component and a positive y-component. A vector in Quad III will have negative x-component and a negative y-component.

  16. y(+) x(-) x(+) y(-) Topic 1.3 ExtendedA - Vector Addition and Subtraction y(+) Standard angles are measured with respect to the +x-axis in a counter-clockwise rotation: θ2 θ1 x(-) x(+) If these angles are used in dx = dcosθ and dy = dsinθ the correct component signs will be automatic. y(-) standard angles Reference angles are measured with respect to either x(+) or x(-), whichever is smaller. θ1 θ2 If these angles are used in dx = dcosθ and dy = dsinθ the correct component signs will NOT be automatic. reference angles Just use the table on the previous page for the signs of the components.

  17. y(+) reference angle x(-) x(+) y(-) Topic 1.3 ExtendedA - Vector Addition and Subtraction y(+) Suppose you have a displacement d of 200-m at 135º. Find its components. 135° x(-) x(+) Using standard angle: dx = dcosθ = 200cos135° = -141.4 m dy = dsinθ = 200sin135° = 141.4 m SIGNS AUTOMATIC standard angle y(-) dy (+) Using reference angle: dx = dcosθ = 200cos45° = 141.4 m dy = dsinθ = 200sin45° = 141.4 m SIGNS NOT AUTOMATIC 45° 180°- 135° = 45° dx (-) dx is (-), and dy is (+) so that dx = -141.4 m dy = 141.4 m Since the signs are not automatic, sketch in components:

  18. Topic 1.3 ExtendedA - Vector Addition and Subtraction Many angles will be given with respect to the points of the compass. NNE Often a compass is divided into 45° increments. ENE Then northeast is precisely 45° between north and east. Sometimes the compass is divided into 22.5° increments. Then we can speak of NNE, and ENE, and assign a precise angle to each.

  19. Topic 1.3 ExtendedA - Vector Addition and Subtraction Finally, we can speak of angles such as "30° east of north," which is a reference to an angle drawn 30° from the north direction, in the eastward direction. north 30° east of north 30° Then you can draw your own, exact, right triangle. 30° 60°

  20. y (ft) y (ft) y (ft) x (ft) x (ft) x (ft) Topic 1.3 ExtendedA - Vector Addition and Subtraction Consider the two vectors A and B shown below. To add the vectors by components, simply add the x-components, then add the y-components. Graphically by components A Ay=2 ft Bx= -1 ft R R Ax=3 ft start end By= -2 ft Graphically B R =A+B A Rx = Ax + Bx = 3 ft + -1 ft B = 2 ft = 2 ft + -2 ft Ry = Ay + By end R start = 0 ft

  21. FYI: In fact, for 2D we only need 2 unit vectors, N and E. Then west is -E and south is -N. Topic 1.3 ExtendedA - Vector Addition and Subtraction A vector can also be expressed without an angle, and without a picture. This method is often preferred, because it requires no pictures, and therefore less paper. But, you can always make sketches if it helps. To facilitate this method we have to define unit vectors. Unit vectors have UNIT LENGTH (meaning a length of 1) and NO QUANTITY. For example, we could use the directions N, S, E, and W as unit vectors, and express any vector in terms of these unit vectors. Thus the displacement D1 of 30 miles to the north would look like this: D1 = 30 miles N Thus the displacement D2 of 40 miles to the west would look like this: D2 = 40 miles W We could also express D2 in terms of the East unit vector: D2 = -40 miles E Then the sum of the vectors D = D1 + D2 can be written D = 30 miles N + -40 miles E

  22. Unit Vectors z y x FYI: Some books use i, j, and k for the unit vectors in the x, y, and z-directions. Topic 1.3 ExtendedA - Vector Addition and Subtraction Naturally, we are not pirates, and so we don't use the point of the compass as unit vectors. Physicists instead use the following: ^ x is the unit vector in the +x-direction ^ z ^ y is the unit vector in the +y-direction ^ z is the unit vector in the +z-direction ^ We read x as "ex hat." (Really...) We rarely need 3D, but if we do, here is the standard configuration of the three axes: ^ y ^ z And here are the unit vectors: Incidentally, they don't have to start at the origin: ^ y ^ x ^ x

  23. y (ft) x (ft) ^ ^ FYI: Your book uses x instead of x. We will use the x notation in this class because it is the standard. Topic 1.3 ExtendedA - Vector Addition and Subtraction To see how the unit vectors work, let's redo the a previous problem where we found R = A + B: ^ y A ^ x R B ^ ^ + 2y (ft) A= 3x ^ ^ + -1x - 2y (ft) B= ^ ^ 2x + 0y (ft) R =

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