Vectors A vector is basically an arrow that represents the magnitude and direction of a measurement. The length of the vector represents its magnitude. Vector A Vector B If vectors A and B represent compatible quantities, i.e., displacement, then the length of the vectors reveals that A is approximately twice as large as B.
The direction of the arrow represents the direction in whatever coordinate system is in use. Vector A is parallel to vector B, meaning they share the same direction. B A A Still parallel. Anti-parallel: meaning 180 degree change in direction. B B
Vector A is perpendicular to vector B, meaning they are at 90 degrees difference in direction. B A B Vector A is oblique to vector B, meaning they are neither parallel, nor perpendicular in relation to each other. A
Notation A vector quantity is usually represented as the variable in question with an arrow over it. Sometime it may be written in boldface rather than with an arrow. The scalar value or magnitude that relates to a vector (i.e., the length) can be represented as the variable of the vector quantity without the arrow (also not in boldface) or as that variable placed inside an absolute value bracket. 𝐴 𝐴 or just A easy to confuse
Coordinate systems (How we write vectors and describe them) Cartesian Coordinates X and Y directions are rectilinear Cartesian space in three dimensions
Cartesian space in two dimensions Axes are interchangeable, and can reference a y-z or x-z plane as needed. We will do most of our work in this course in 2D as once these principles are mastered, adding the third dimension is more of the same as far as the mathematics are concerned.
Remember that when referencing the Cartesian system you say: “Vertical axis” vs. “Horizontal axis.” For a standard x-y plane, your function would be: “y vs. x,” or “y with respect to x.”
Cartesian Coordinates We can represent a vector in Cartesian coordinates by giving its end point (the tip of the arrow) as an ordered pair (2D) or ordered triple (3D) (x,y) (x,y,z) (3,4) (3,4,5) Sketch These! This notation is O.K., but less meaningful in a physics context (my opinion). The idea of representing a vector this way is, however, VERY useful in computer science and graphic animation.
Now, another method… We can represent a vector in Cartesian coordinates by showing the Resultant as the sum of x,y, and z component unit vectors. A unit vector is simply a vector of magnitude (length) 1 unit in a given direction. In this case, the x-, y-, and z-directions, respectively. Sometimes instead of x,y,z we instead use i,j,k to represent the same thing.
This looks like: Let’s use a vector representing a force acting on an object, say the force of the wind on a kite. If the wind was blowing with a force of 10 Newton’s (unit of force) North and 6 N to the East we might write any of these. 6 𝑁 𝐸𝑎𝑠𝑡+10 𝑁 𝑁𝑜𝑟𝑡ℎ 6 𝑁 𝐸 +10 𝑁 𝑁 6 𝑁 𝑥 +10 𝑁 𝑦 Best
𝑅 = 6 𝑁 𝑥 +10 𝑁 𝑦 R Notice how the vector is expressed as the sum of force values multiplied by a unit vector (literally a length of 1-unit in the given direction)? This is because the vector itself is, in fact, the sum of these components, an element we will use to our advantage in a later lesson. 10 N 6 N
Relate a radius (the magnitude of the vector) Polar Coordinates Relate a radius (the magnitude of the vector) and an angle of incline Angles are always designated from the positive x-axis (or East, if you use the cardinal points) unless otherwise indicated. This convention is routinely employed in math and science, but not in navigation.
(r,) (12 m, 30o) 12 m @ 30o 12 m @ 30o N of E 12 m @ 60o E of N -12 m @ 30o S of W -12 m @ 210o Draw these vectors. 12 m 300 These are all the same vector!
Practice Sketch the following vectors. 1) R = -3x + y 2) R = 10x 3) R = 3y 4) R = 2y + 6z 5) R = i + 3j 6) R = 2j – 4i R = 4i – 2j R = x + 2y + 3z
Practice (cont’d) Sketch the following vectors 1) 30 m/s @ 15o N of E 2) 12 m E 3) 9.8 m/s2 down 4) 50 m/s @ 130o 5) 50 m/s @ 50o N of W 6) -13 m @ 10o E of S 7) 22 m/s @ -20o 9) 22 m/s @ 20o 10) r = 6m @ 35o 11) r = 3m @ 270o 12) r = -9m
Tools: Pythagorean Theorm
Resolution of independent vector components As we have seen with our Cartesian notation, a vector can be represented as the sum of its parts. If we let the hypotenuse of a right triangle represent the a vector, the legs of that triangle represent the horizontal and vertical components of that vector. This allows us to break a vector down to find out its magnitude in the horizontal (x) and vertical (y) directions. Why do you think this would be important?
Why might it be important in physics to disassemble a vector into its horizontal and vertical components?
Answer: Because in physics we can utilize the principles of linear motion to then analyze each part, or component of the motion independently of the other.
We use this technique in physics because, as you will learn shortly, vectors of the same variable that are at right angles to each other do not have any effect on each other. That means that motion in the horizontal direction does not have any effect on motion in the vertical direction! While we make that statement, it remains important to remember that the path of an object (its displacement) depends on both components.
Unfortunately... We need trigonometry to do this... Side a corresponds to angle A Side b corresponds to angle B Side c corresponds to angle C
sin = opp/hyp cos = adj/hyp tan = opp/adj sin A = a/c cos A = b/c tan A = a/b sin B = b/c cos B = a/c tan B = b/a So if you know any two angles, and any two sides you can extrapolate the rest of the triangle.
Example Rx = Rcoso Rx = 50cos20o Ry= Rsino Ry = 50sin20o If a car travels at 50 m/s at 20o North of East, find the horizontal(east) and vertical (north) components of the velocity. Rx = Rcoso Rx = 50cos20o Ry= Rsino Ry = 50sin20o
BE CAREFUL!!! You will come to notice that we often use sine for your “y” component and cosine for your “x” component, but it always depends on the orientation of the given angle within the system. Write out your trig defn’s and carefully examine your figures EVERY TIME.
We won’t use this one much in class, but it is incredibly useful in general geometry, particularly on SAT and ACT math. As a bonus it is easy to remember. The Law of Sines This relationship will allow you to solve ANY triangle long as you know at least 1 side and 2 angles, or 2 sides and 1 angle. This is very handy for right triangles since you always know at least 1 angle (90o) and have the Pythagorean theorem available.
Adding vectors (a preview of next lesson) When adding vectors, place the 1st vector at the origin. Next place the tail end of one vector to the head end of the other. Then draw the resultant vector from the origin to the tip of the second vector. 10 N 6 N R Hey that looks like the representation of a Cartesian vector!
More practice: Relative velocities 1) Find the resultant velocity of a boat that crosses a river due east @ 4 m/s while the current runs south @ 1 m/s. 2) What is the displacement of a plane that flies south for 3.0 hours at 500 km/h with a 20 km/h tailwind? A 15 km/h headwind? 3) A cannonball is shot upwards at an angle of 30o above the horizontal with a velocity of 35 m/s. Find the horizontal and vertical components of the velocity. Draw these component vectors. 4) A car drives down a street at 30 m/s. A man is walking in the same direction as the car at 2 m/s as he passes a stationary mailbox. What is the velocity of the car with respect to the man? The car with respect to the mailbox? The man with respect to the car?
4) An evil physics student fires a potato gun forward out of a truck traveling at 25 m/s. If the gun propels the potato at 52 m/s, how fast is the potato traveling When it strikes a stationary parked car? When it strikes a cyclist riding forward (in the same direction) at 4 m/s? When it strikes a cyclist riding “backwards” relative to the truck at 4 m/s? Now the potato was fired backwards off of the truck. Do a), b) & c) for this case.
5) Train A heads east at 175 m/s. Train B heads west at 150 m/s 5) Train A heads east at 175 m/s. Train B heads west at 150 m/s. What is the velocity of train A with respect to train B? 6) The trains in problem 5 are now both traveling north. What is the velocity of train B with respect to train A? What is the velocity of train A with respect to train B?