1. Scalars A scalar is anything that can be represented by a number (+ or -) and any associated units. A scalar is always written with a non-bold symbol and can be positive or negative. The magnitude of a scalar x is the absolute value |x| and is always positive. A vector has both a magnitude and a direction. Vectors Vectors and Scalars

2. You can use boldface and larger font size, e.g. X is a vector. how to write a vector You can use the letter that symbolizes the vector with an arrow above it: x This is also a vector: You don’t need to make the symbol boldface or put an arrow above it (not wrong, but why do the extra work?). The length of the arrow represents the vector’s magnitude and the letter symbol is its label.

3. The letter symbol alone is the vector’s magnitude: A = |A|. Magnitudes are always positive! A vector can never equal a scalar. Never write A = A or. Repeat after me: “Magnitudes are always positive.” Repeat after me: “A vector can never equal a scalar.”

4. Remember, vectors have both magnitude and direction. You can specify a vector by:  magnitude and direction  magnitude and angle it makes with some axis  components with respect to axes. duh!

5. How to lose points and reduce your Physics 35 grade:  forget to specify the direction of a vector  set a vector equal to a scalar (similar to above)  write down a negative value for a magnitude  calculate the magnitude of a vector by adding the magnitudes of its components

6. Vector Components Here’s a vector: 99.9% of the time you need to specify coordinate axes. Any vector not along an axis needs to be resolved into its components. Use a dotted line, or a dashed line, or a less- thick line to distinguish components from the vector itself. x y A You should not assign separate labels to the components. Why?Why? You don’t need to. They already have a name shown. Extra labels could lead to confusion and mistakes.

7. x y Technicalities! This entity is a vector! (An arrow with a label.) A AxAx AyAy So is this! (An arrow with a label.) Those things with the labels and subscripts are the vector components of the vector A. Vector components are themselves vectors. I said don’t show the labels in your figure, so let’s get rid of them.

8. x y A In Physics 23, the dashed arrow things aren’t vectors. They are just components. “Stop! You’re confusing me!” I know it’s confusing, so let’s do an example. Suppose there is a mass M being acted on by force A. Calculate the x- and y- components of the acceleration of the mass. Let’s call the angle that the force makes with the x-axis .  Components are the + or – entities that make up a vector. “Vector component” implies you are thinking of the component as a vector by itself. This is nitpicky terminology that physicists get sloppy with and is not worth your worrying about.

9. x y A M OSE: Remember the rule: component versions of OSE are automatically OSE.  Let’s do the x-component first. Use your fingers! Always leave expressed in component form if you are unsure of their direction. You’ll need to attend class to see this in action. a

10. x y A M  a For this problem, I have to give the answer in component form, because that’s what I was told to do. y-component next

11. Quiz from Winter 2003. “We” were not quite prepared for force problems! What are the forces that act on the mass? Normal force? A force up the plane? A “velocity” force? Normal force? NOT THAT ONE! A force up the plane? NO! A “velocity” force? NO! θ v M Sample Vector Application A mass is released on a frictionless inclined plane. Calculate the force on the mass by the plane and the acceleration of the mass.

12. θ The only forces are the weight “down” and the normal force perpendicular to the plane’s surface. We are interested in the motion of the block. Only include forces on the block. Do not include forces on the plane! w=Mg The block will accelerate down the plane. Acceleration is NOT a force. Do not “attach” it to the block! N a You can choose any axes you want. If you choose axes like this, the acceleration has both x- and y- components. You will have to do twice the work to find a. Twice the opportunity for errors. Do you really want to do that? M

13. θ w=Mg N a It is a good idea to choose one axis parallel to the acceleration! y x Resolve any forces not parallel to an axis into their components along the chosen axes. θ Choose an OSE. Note that the acceleration is parallel to the x- axis. You may not need to sum the forces in the y-direction. Don’t do it unless you need to. The LHS is the sum of forces. There are two forces on the mass. I expect you to include all of the forces! You may choose not to, but I promise I will take points off! M

14. θ w=Mg N a y x θ M 0 The component of the weight parallel to the plane is the force that accelerates the mass down the plane. You know the direction of acceleration (because I told it to you in the statement of the problem). So do not leave it as a component! Now do the algebra! Next: normal force.

15. θ w=Mg N a y x θ M 0

16. One more thing. The symbols we use talk to you. Listen to them. This says “I am a vector. I have both a magnitude and a direction. You have to specify both! Never set me equal to a scalar (if you do, bad things will happen).” Q Q (If Q is a vector) This says “I am the magnitude of the vector Q. I am always a positive number. Never set me equal to a negative number. Never set me equal to a vector (if you do, I promise you will have physics nightmares).”

17. (If z is not a vector) This says “I am a scalar. I have a numerical value, and possibly units and a negative sign buried inside me. Never set me equal to a vector (if you do, your inbox will be filled with 100 100 spams).” z PyPy This says: I am the y-component of the vector P. I am not myself a vector, but I do have a magnitude, and I do have a sign. To get an answer correct, you have to specify both. If you don’t, Dr. Pringle will make fun of you in class.” Remember: to specify a vector, you must specify both its magnitude and direction (“how much, and which way”). To specify a vector component, you must also specify how much and which way, but “which way” means pick a + or – sign. If you write the symbol E x, you are telling me that you haven’t yet determined which sign!

18. “This vector component stuff is really confusing and you are being really nitpicky and I’ll never get it!” I agree it is confusing. Yes, I am being nitpicky. Physicists can be very casual about expressing themselves. If we know in our head what we mean, who cares what it looks like on paper, right? Besides, when we talk with other physicists, we all think alike, so all other physicists know what we meant, even if we didn’t write it that way, right? Find a way to convince me that you at least partway get it!