2. Topic 1.3 Extended A - Vector Addition and Subtraction

3.  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. Topic 1.3 Extended A - Vector Addition and Subtraction  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.

4. Topic 1.3 Extended A - 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.  Instead, we need an arrow to show its direction. Such an arrow is called a vector.  A vector has both magnitude (numerical size), and direction.  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 y x y x y vector

5. Topic 1.3 Extended A - Vector Addition and Subtraction  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:  Note that the displacement vector (black) does not necessarily show the actual path followed (purple).  A displacement is simply a directed change in position, with disregard to the route taken. displacement vector actual path Note that the displacement vector traces out the SHORTEST distance between two points. FYI: The displacement from Sheboygan to Milwaukee has the same magnitude. What is different about it?

6. Topic 1.3 Extended A - 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 = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 distance formula √ √ √ √ √ √  For example, the displacement shown has a magnitude given by d = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 = (5 - 1 ) 2 + (4 - 1 ) 2 = 4 2 + 3 2 = 16 + 9 = 25 = 5 ft. x (ft) y (ft) (x 1, y 1 ) = (1, 1) (x 2, y 2 ) = (5, 4) start end

7. Topic 1.3 Extended A - Vector Addition and Subtraction  Of course, a scale drawing would allow you to measure the magnitude of the displacement directly with a ruler: x (ft) y (ft) (x 1, y 1 ) = (1, 1) (x 2, y 2 ) = (5, 4) start end  Note that the length of the arrow is 5 ft.

8. Topic 1.3 Extended A - Vector Addition and Subtraction  In print, vectors are designated in bold non-italicized print.  Consider two vectors: vector a and vector b.  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). a b tail tip

9. Topic 1.3 Extended A - 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.  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. We are giving the sum an arbitrary name - say s. a b tail tip "translate" means to move without rotation. s tail tip

10. Topic 1.3 Extended A - 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!

11. Topic 1.3 Extended A - 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. a b start end a + b = s s  The vector s represents the shortest distance to the treasure.

12. Topic 1.3 Extended A - Vector Addition and Subtraction  Now, suppose we want to subtract vectors. b -b-b the vector b the opposite of the vector b a -b-b a+-ba+-b a-b = a + -b-b Thus,  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.

13. Topic 1.3 Extended A - Vector Addition and Subtraction  Observe the addition of vectors is commutative: a + b = b + a c b a (a+b)(a+b) c b a (a+b)+c (b+c)(b+c) a+(b+c)  Observe that addition of vectors is associative: (a + b) + c = a + (b + c) b a a b a+ba+b b+ab+a

14. Topic 1.3 Extended A - 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. (b) How far does the pirate walk? (c) Draw in the displacement vector. (d) What is the magnitude of the displacement? N (pc) E (pc) 6 paces 4 paces 2 paces 1 pace 6 paces 4 paces2 paces1 pace + + + = 13 paces 5 paces

15. Topic 1.3 Extended A - Vector Addition and Subtraction  Suppose you know a vector's magnitude and direction. opp hyp adj hyp opp adj hypotenuse adjacent opposite θ trigonometric ratios s-o-h-c-a-h-t-o-a θ d d x = d cos θ d y = d sin θ d dxdx dydy d d d x = d cos θ d y = d sin θ sin θ = cos θ = tan θ = dxdx dydy  You can find its components using the trigonometric functions:

16. Topic 1.3 Extended A - Vector Addition and Subtraction  Recall the four quadrants of the Cartesian coordinate system: x(+) x(-) y(+) y(-) I II III IV I++I++ II-+ III-- IV+- Quadrantxy  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.

17. Topic 1.3 Extended A - Vector Addition and Subtraction  Standard angles are measured with respect to the +x-axis in a counter-clockwise rotation:  If these angles are used in d x = d cos θ and d y = d sin θ the correct component signs will be automatic.  Reference angles are measured with respect to either x(+) or x(-), whichever is smaller.  If these angles are used in d x = d cos θ and d y = d sin θ the correct component signs will NOT be automatic.  Just use the table on the previous page for the signs of the components. x(+) x(-) y(+) y(-) θ1θ1 θ2θ2 standard angles x(+) x(-) y(+) y(-) θ1θ1 θ2θ2 reference angles

18. Topic 1.3 Extended A - Vector Addition and Subtraction 45°  Suppose you have a displacement d of 200-m at 135º. Find its components.  Using standard angle: d x = d cos θ = 200 cos 135° = - 141.4 m d y = d sin θ = 200 sin 135° = 141.4 m SIGNS AUTOMATIC  Using reference angle: d x = d cos θ = 200 cos 45° = 141.4 m d y = d sin θ = 200 sin 45° = 141.4 m SIGNS NOT AUTOMATIC d x is (-), and d y is (+) so that d x = - 141.4 m d y = 141.4 m x(+) x(-) y(+) y(-) 135° standard angle x(+) x(-) y(+) y(-) reference angle 180 ° - 135° = 45 ° d x (-) d y (+) Since the signs are not automatic, sketch in components:

19. Topic 1.3 Extended A - Vector Addition and Subtraction  Many angles will be given with respect to the points of the compass.  Often a compass is divided into 45° increments.  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. ENE NNE

20. Topic 1.3 Extended A - 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. 30° 30° east of north north  Then you can draw your own, exact, right triangle. 30° 60°

21. Topic 1.3 Extended A - Vector Addition and Subtraction  Consider the two vectors A and B shown below. y (ft) x (ft) A B  To add the vectors by components, simply add the x-components, then add the y-components. A x = 3 ft A y = 2 ft B x = - 1 ft B y = - 2 ft R = A + B R x = A x + B x = 3 ft + - 1 ft = 2 ft R y = A y + B y = 2 ft + - 2 ft = 0 ft y (ft) x (ft) Graphically by components R start end R y (ft) x (ft) start end A B R Graphically

22. Topic 1.3 Extended A - 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 D 1 of 30 miles to the north would look like this: D 1 = 30 miles N  Thus the displacement D 2 of 40 miles to the west would look like this: D 2 = 40 miles W  We could also express D 2 in terms of the East unit vector: D 2 = -40 miles E FYI: In fact, for 2D we only need 2 unit vectors, N and E. Then west is -E and south is -N.  Then the sum of the vectors D = D 1 + D 2 can be written D = 30 miles N + -40 miles E

23. ^ Topic 1.3 Extended A - 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 ^ y is the unit vector in the +y-direction ^ z is the unit vector in the +z-direction ^ Unit Vectors FYI: Some books use i, j, and k for the unit vectors in the x, y, and z- directions.  We read x as "ex hat." (Really...) ^ x y z  We rarely need 3D, but if we do, here is the standard configuration of the three axes:  And here are the unit vectors: x ^ y ^ z  Incidentally, they don't have to start at the origin: x ^ ^ y ^ z

24. Topic 1.3 Extended A - 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 (ft) x (ft) A B A =A = R 3x3x ^ x ^ y ^ + 2y (ft) ^ B =B = -1x ^ - 2y (ft) ^ R = 2x ^ + 0y (ft) ^ + FYI: Your book uses x instead of x. We will use the x notation in this class because it is the standard. ^ ^