Vectors. Vector Practice Problems: Odd-numbered problems from

Vectors Vector Practice Problems: Odd-numbered problems from 3.1-3.21 After today, you should be able to: Understand vector notation Use basic trigonometry in order to find the x and y components of a vector (only right triangles) Add and subtract vectors

Reminder: Scalars and Vectors Scalar: Just a number. Vector: A number (magnitude) with a direction. I have continually asked you, which way are the v and a vectors pointing? a v Today we ll be all about vectors and I want to remind you of terms from lecture 3. Also want to remind you that these are just official-sounding terms for a simple concept. Here are the definitions. Direction can be east, up, positive, negative. I ve been asking you things like tell me which way the a vector is pointing, although we ve been drawing vectors with arbitrary length. Today we re going to properly discuss vectors, their properties, components, and basic mathematics. Much of the first part of this lecture is going to be a bit abstract so I wanted to start with a physical anchor point, and give you this problem to set up why we re doing this abstract math.

Vectors A car drives 50 miles east and 30 miles north. What is the displacement of the car from its starting point? North 30 miles North 50 miles East East Displacement is a vector we re familiar with, but now it s going in two dimensions. You can see that it s got a horizontal component and a vertical component. This final displacement vector is defined by a MAGNITUDE and DIRECTION.

Vectors A car drives 50 miles east and 30 miles north. What is the displacement of the car from its starting point? North 30 miles North Displacement is a vector (net change in position) 50 miles East East Displacement is a vector we re familiar with, but now it s going in two dimensions. You can see that it s got a horizontal component and a vertical component. This final displacement vector is defined by a MAGNITUDE and DIRECTION.

Describing a vector A vector is described *completely* by two quantities: magnitude (How long is the arrow?) & direction (What direction is the arrow pointing?) A vector is described *completely* by two quantities, the magnitude and direction. On a graph, a vector is represented by an arrow on the y/x graph. The arrow indicates direction and length indicates magnitude.

Magnitude and direction North Magnitude: length of this line θ 50 miles East 30 miles North Direction: angle from reference point (here, θ degrees North of East ) East When we think back to the car can see magnitude and direction of this arrow as we might see it and state it. It s got a vector, and we can describe this as θ degrees Northward from the East direction. Now you ll note that this is a right triangle. So you can see already how trigonometry will come into play here: we can figure out the properties of all vectors in this course by breaking things down into right triangles. We ll do this a little later.

Vector notation c d This vector written down: A A cd And its magnitude A A cd Now let s go through basic vector properties. I ve labelled this to show you how you ll see vectors written. I ve called this vector A. You can see it can be written as arrow-hat, or just bold. The book writes it in the way I ve shown here, with the arrow-hat. You may also see vectors with points labelled c and d as the start/end of the vector, although we won t do that.

Vector components Ay Ay Vector change in y direction Ax Ax Vector change in x direction Now let s go through basic vector properties. When you re in a coordinate system, like y/x, Vectors can be described by their independent components, the vector s change in the vertical axis and the change in the horizontal axis. The notation for these is putting a subscript with the relevant axis name. So like Ay was our 30 miles north and Ax was our 50 miles east in the driving example.

Basic vector operations Translating vectors Vectors are defined by ONLY magnitude and direction. = = = These are all the SAME vector! Let s think about what a vector means: it s described completely by its length and direction. NOT ITS LOCATION ON THE AXES! When doing vector math you can slide a vector all around here and it s not going to change its properties, because it s not going to change its magnitude and direction. In that way, all these vectors can be written as EQUAL because they have the same length and direction.

Basic vector operations Multiplying by -1 V, has an equal magnitude but opposite direction to V. = Now let s think about multiplying a vector by -1. If you multiply a vector by -1, all that happens is that it makes the DIRECTION OPPOSITE. If you multiply A by -1, it swaps direction. Minus one inverts the direction of a vector.

In which case does =? A. B. C. D. Q17 Answer: B. Remember: Vectors are defined by MAGNITUDE AND DIRECTION! A = -B means B has opposite direction but same magnitude!

Basic vector operations Geometrically adding vectors [m/s] [m/s] Two vectors with the SAME UNITS can be added. Let s talk about adding vectors together. First and most importantly! As with any addition, if you re adding two vectors they MUST HAVE THE SAME UNITS! So here are two vectors - can they be added? I ve written their units there, and we can see they re both velocity vectors because they re in m/s. So they can be added.

Basic vector operations Geometrically adding vectors + =? tail tip So what does it mean to ADD a vector quantity. Importantly, as I noted before you can always do what we call translating a vector. We do this here because when you re adding vectors, they must always be summed tip to tail.

Basic vector operations Geometrically adding vectors + =? tail tip When adding geometrically, always add tail to tip! So what does it mean to ADD a vector quantity. Importantly, as I noted before you can always do what we call translating a vector. We do this here because when you re adding vectors, they must always be summed tip to tail.

Basic vector operations Geometrically adding vectors + = vector + vector = vector This is called the triangle method of addition The resultant vector can be drawn from the start of the first summed vector to the end of the last summed vector B. Now a few things to note here: 1. Adding summed components tail to tip. 2. Final vector drawn from start point to end point. 3. Final result of adding two vectors is A VECTOR. Term for this type of addition where you translate vectors to sum them is the triangle method of addition

Basic vector operations Geometrically adding vectors + = + = It s commutative! It doesn t matter which one you add first. You can start which withever you like. If you add them correctly you will end up in the same place.

If you were to add these two vectors, roughly what direction would your result point? Q18 A. B. C. D. E. None of the above Answer: C. Use triangle addition, put tail to tip.

V1 + V2 = VR Translate the vector and always add tail to tip!

What is + =? Q19 A. B. C. D. E. None of the above Answer: C. Use triangle addition, put tail to tip.

Basic vector operations Geometrically subtracting vectors - =? When adding/subtracting geometrically, always add tail to tip! What does it mean to subtract a vector? We want to subtract B from A.

Basic vector operations Geometrically subtracting vectors - - = + (- ) When adding/subtracting geometrically, always add tail to tip! Let s break this equation down and note that subtracting B is the same thing as adding negative B. So we take the negative of B, which makes the direction of B opposite to what it was. Now JUST AS BEFORE, we want to translate thigs, and SUM TIP TO TAIL. So let s line these up right (click), pause, (click)

Basic vector operations Geometrically subtracting vectors - - = + (- ) When adding/subtracting geometrically, always add tail to tip! And then draw the resultant vector from the starting point of the first to the end point of the second.

Vectors A car drives 50 miles east and 30 miles north. What is the displacement of the car from its starting point? North 30 miles North 50 miles East East Lest this become too abstract, I wanted to link vector addition/subtraction back to our concrete problem, where we were just driving along a line. remember we can add, subtract, find magnitudes of vectors just as if we were driving along the line (tail-to-tip).

North Vectors A car drives 50 miles east and 30 miles north, then 20 miles south. What is the displacement of the car from its starting point? 20 miles South East Here we can add a southern part of the trip, where we re then going along the negative axis: this is like subtracting this last vector. Graphically, I like to think of vector addition/subtraction as making sure you re always following the flow of the path.

R Fig. 3.4 in your book In graphical addition/subtraction, the arrows should always follow on from one another, and the resultant vector should always go from the starting point to the destination point in your summed vector path.

Basic vector operations Scalar multiplication 3-3 Multiplying a vector A and a scalar (i.e. number) k makes a vector, denoted by ka. Last vector math thing before we start applying this concept. Multiplying by a number changes the magnitude.

Vector arithmetic: components North What is D (the magnitude of )? 50 miles East Dx 30 miles North Dy East A. 58 miles B. 80 miles C. 20 miles D. 0 miles E. 58 m/s Q20 Answer: A. I want you to think about your trigonometry and answer this question.

Vector arithmetic: components North What is D (the magnitude of )? 50 miles East Dx 30 miles North Dy East A. 58 miles B. 80 miles C. 20 miles D. 0 miles E. 58 m/s Q20 Think about the Pythagorean Theorem Answer: A. I want you to think about your trigonometry and answer this question.

Vector arithmetic: components A Ay Ax A, Ay, and Ax here are the MAGNITUDES of the vectors drawn (they don t have hats and are not bold). Earlier I pointed out that our car s path made a right triangle, and that trigonometry would come into play for vector addition. Let s lay that out now. Here s a representation of a vector A; Ax is the x component, Ay is the y component. I note here that I ve written these as MAGNITUDES; you can see it in the notation. This illustration is simply to show: the vector components form a right triangle with the final vector.

Vector arithmetic: components A Ay Ax The magnitude of a vector component is its final number minus initial number! I also want to point out something that I know will trip you guys up Describe this in terms of length of the vertical and horizontal arrows. THE SIZE OF THE X COMPONENT OF A IS ITS LENGTH, so X FINAL MINUS X INITIAL. SAME GOES FOR Y!

Vector arithmetic: components yf A Ay Ay = Ay = yf - yi xi Ax yi xf Ax = Ax = xf - xi Here is that statement in formal math notation.

Vector arithmetic: components North What is the magnitude of the x component of? D D 70 miles 30 o Dx Dy East A. 60.6 miles B. 35.0 miles C. 40.4 miles D. 0 miles E. 31 degrees Q21 A is the right answer (use cosine)! B if took sin() C if took tangent() When you re asked about vector components, think on your feet! What do you know and what could you find out? DRAW ON LIGHT BOARD! I want you to think about your trigonometry and answer this question.

Vector arithmetic: components North What is the magnitude of the x component of? D D 70 miles 30 o Dx Dy East A. 60.6 miles B. 35.0 miles C. 40.4 miles D. 0 miles E. 31 degrees Q21 Think about SOH CAH TOA! A is the right answer (use cosine)! B if took sin() C if took tangent() When you re asked about vector components, think on your feet! What do you know and what could you find out? DRAW ON LIGHT BOARD! I want you to think about your trigonometry and answer this question.

Using trigonometry, you can find all vector components and angles given just a bit of information! Look at the triangles, and think about what you can figure out based on available info.

Ultimate rule of vector math Don t fear the vector. To study: Practice drawing/graphing vector operations. Get used to vector and magnitude notations. Practice solving for x, y components. Practice solving for θ.

Diandra kicks a soccer ball to a max height of 5.4 m at a 20 angle from the ground with a speed of 30 m/s. What is the x (horizontal) component of the initial velocity of the soccer ball? There s a little more time so I ll set you up a problem for next week If there s time give this as a walk-around example. If no time, start with this one next week.

Diandra kicks a soccer ball to a max height of 5.4 m at a 20 angle from the ground with a speed of 30 m/s. What is the x (horizontal) component of the initial velocity of the soccer ball? TO START: Draw your vector right triangle. What are the sides? Compare your triangle with your neighbor. There s a little more time so I ll set you up a problem for next week If there s time give this as a walk-around example. If no time, start with this one next week.

CAREFUL! You can t do vector arithmetic combining displacement (5.4m) with speed (30m/s)! 30 m/s 5.4 m 20 THIS TRIANGLE IS WRONG BECAUSE 5.4 M IS NOT THE Y COMPONENT AT ALL!!! Balls will travel in a parabolic motion and the peak of that parabola is 5.4m high. ALSO, THE UNITS ARE WRONG! You want to solve for the x component of the VELOCITY VECTOR. cos (20 deg) = V_x/(30 m/s)