vectors...
what means to "decompose or resolve" a vector?
"broke" a vector into two perpendicular vectors that are parallel to the coordinate axes (component vectors)
How displacement is symbolized?
(triangle, which means change) r (arrow above)
Name the notation used with the displacement (vector name, vector components, components, magnitude)
-Displacement: (triangle) r (arrow above r) -Vector components: (triangle) x, (triangle) y (arrow above letters) -Components: (triangle) x, (triangle) y -Magnitude: (triangle) r
examples of scalars? (4)
-length -speed -mass -temperature
Scalar components (other name)
-or simply components of vector V
Z is ___ in the XY plane, X is ____ in the YZ plane, Y is ____ in the XZ plane
0: (x,y,0) 0: (0,y,z) 0: (x,0,z)
What is the unit vector of vector a and how is it found?
A unit vector is a vector with the same direction as a but with a magnitude of 1. It is found by dividing the components of a by the magnitude of a. This gives you components that add up to a length/magnitude of 1.
What is the dot product? How is it denoted?
Also known as the scalar product. It is a scalar not a vector. Essentially it is vector multiplication with the multiplication in different dimensions. It can also relate the angle between two vectors. Denoted A · B
which type of coordinate system will you use for physics problems?
Cartesian coordinate system
How is the dot product found given two vectors and their components?
Given two vectors with components a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃) the dot product is: a₁b₁ + a₂b₂ + a₃b₃
When are vectors equal?
If they have the same magnitude AND direction
Scalar Vector projection of a onto b? How is this denoted and how is it found?
Length of the vector projection of a onto b. Or rather the magnitude of the projection as it is not a vector. comp sub a b= (a · b) / ||b||
What does it mean to be orthogonal?
Perpendicular.
How are vectors added and subtracted ALGEBRAICALLY?
Simply add/subtract their corresponding x components and y components to obtain the resultant vector.
How is the angle between two vectors found using the cross product?
Solve for theta.
What is an alternative way to find if TWO vectors are parallel using the dot product?
Solving for theta. If you have an angle of 0 or 180 degrees between the vectors then you have a straight line in either case. Therefore, they are parallel.
What are the direction angles of a vector a? How are they found?
The angles α, β, and γ, on the interval [0, π] that the vector makes with the x, y and z axis. Take the unit vector of a and take cos⁻¹ of each component to get the angle to find.
If A x B = 0 then?
The vectors are parallel.
Standard basis vectors?
These are the ith, jth and kth vectors. i = <1, 0, 0> j = <0, 1, 0> k = <0, 0, 1>
How are vectors added and subtracted GRAPHICALLY? How is the resultant drawn?
To subtract just attach tail of the negative of B to the head of A. Resultant is drawn by attaching the tail to the tail of A and the head to the head of B
How are the x and y components found given the magnitude and the angle from the origin?
Treat it like a trig problem. This is simply a triangle with the hypotenuse (r) and 2 angles given (theta and 90°) So: Vx = cosθ * r Vy = sin θ * r
How can we perform subtraction between vectors?
With numbers, subtraction is the same as addition of a negative number ex: A-B=A+(-B)
cosine
adjacent/ hypotenuse
solve cosine for adjacent
adjacent= h cos (delta)
To add two vectors the need to be arranged_
head to tail
x-component vector
the vector (projection) along the x-axis
y-component vector
the vector (projection) along the y-axis
the negative of a vector A, is defined to be _
the vector of the same length as A, pointing in opposite direction
Vector components
the velocity vector may be replaced by the sum of the 2 vectors, one of which is horizontal (Vx) and the other which is vertical (Vy). Vx and Vy are both vectors, so are called the vectors components of the vector V
How does the dot product tell us if two vectors are perpendicular to each other?
to be perpendicular would mean there is a 90° between the two vectors. Cos 90⁰ = 0 and therefore if the dot product between vectors is 0 they are perpendicular. Basically if the dot product is 0.
The definition of acceleration involves the difference between_
two instantaneous velocities
what are the rules of multiplication of vectors by scalar numbers?
-Multiplying a vector by a positive scalar gives another vectors of different magnitude but pointing in the same direction -When a vector is multiplied by a negative scalar, the resultant vector (negative value vector) must be such that, when it is added to the original vector, the resultant is the zero vector. In other words, the tip of vector -A must return to the tail of vector A. All this without changing the length of the negative vector
Name the notation used with position (vector name, vector components, components, magnitude)
-Position: r (with arrow above the letter) -Vector components: x, y (with arrow above the letter) -Components: x, y (no arrow) -Magnitude: r (no arrow)
Name the notation used with the Vector A (vector, vector components, components, magnitude)
-Vector: A (with an arrow above the letter) -Vector components: Ax, Ay (with an arrow above the letter) -Components: Ax, Ay (no arrow) -Magnitude: A (no arrow
Vx can only point in which directions? Vy Can only point in which direction? Explain why we use positive and negative signs with the component vectors
-Vx can only point to left or right -Vy can only point up or down -Because of this, it makes sense to use a positive or negative sign to designate the direction of these two vectors
The direction of a diagonal vector is indicated by an_. Also by _
-angle measured relative to coordinate axes placed at the tail of the vector -describing the vector by an angle measured counter-clockwise from the +x-axis (which will lack the qualifier of north, east, west, south, etc)
examples of vectors? (5)
-displacement -velocity -acceleration -force -electromagnetic fields
net displacement is the vector
-from the initial position to final position -The addition from the two complement vectors that forms it (Vx and Vy)
to describe a vector, what 2 things we must specify?
-magnitude -direction
vector addition
-mathematical operation combining 2 vectors A and B to form a third resultant vector R, the "vector sum" of A and B
the units of the magnitude of a vector could be _(3 examples)depending on which particular _
-meters -meters/second -meters/second² -quantity each vector represents
does vectors have sign? why?
-no -the words "positive" and "negative" would only make sense if there were only 2 possible directions. A vector can point into an infinite number of directions
The magnitude of a vector, a_quantity, cannot be_
-scalar quantity -cannot be a negative number
How we assign the positive and negative to the component vectors?
-the standard coordinate axes are assumed -Positive to right or up -Negative to left or down
zero vector
-vector multiplied by a scalar of quantity zero that has 0 length or magnitude, so its direction is irrelevant
component vectors
-vectors that are parallel to the axis of the Cartesian coordinate system -both its length and direction are determined by specifying the coordinates of the point at the tip of the vector; these coordinates are called the components of the vector
MISSING LABS Lab 11: Vectors and Particles - Inv. #2 Lab 12: Vectors and Particles - Inv. #3 Lab 13: Vectors and Particles - Ex.
...
A plane is in _____ and a space is in _____
2D/an ordered pair of real numbers 3D/an ordered triple of real numbers
If a vector starts at (x₁, y₁) and ends at point (x₂, y₂) then its components are ____?
<x₂ - x₁ , y₂-y₁>
Position Vector?
A vector with its initial point at the origin of the graph. A = {0,0,0} B = {a₁,a₂, a₃} Vector AB is thus simplified to: {a₁,a₂,a₃}
What is the cross product? How is it denoted?
Another way to take the product of two vectors. It differs from the dot product in that it must be in R³ and produces a vector not a scalar. The vector is orthogonal to A and B. Also known as the vector product. Denoted: A x B
How is the magnitude of a vector found?
Apply the pythagorean theorem to the x and y components. The hypotenuse you solve for is the magnitude. Or rather, apply the distance formula to your components.
VECTOR projection of a onto b? How is this denoted and how is it found?
Essentially the shadow of vector a on to b. Or rather how far along the direction of b does a go?
scalar quantities are ones described by_
a magnitude only
In graphical form, vector quantities are represented by_
arrows
How vectors are symbolized?
by a letter with a small arrow above it (small arrow does not point in the direction of the vector; it merely symbolize that the vector by a quantity has a direction associated with it)
A vector may be described quantitatively in 2 different ways:
by describing its magnitude (size) and direction or describing its horizontal and vertical components
What is the second method to describe a vector?
by using its horizontal and vertical components
If the angle is stated without a qualifier, the angle is assumed to be measured _
counterclockwise from the +x-axis (clockwise if the angle is negative)
solve equation for delta
delta= tan⁻¹ (opposite/adjacent)
The velocity vector has both _
direction and magnitude (size)
define displacement of an object during an interval of time (include other name for this)
displacement of an object during an interval of time (also called change in position of the object) is defined as the difference between the position of the object at the end of the time interval (rf (arrow above)) and the position at the beginning of the time interval (ri (arrow above))
To draw a diagram that correctly illustrates the sum of the two vectors you must both_
draw arrowheads on each vector label each vector
scalar multiples _____ a line but can't ___
extend or shorten (extend if c > 1, shorten c <1) get off the line/change the direction to anything but the opposite direction
write expression of hypotenuse in terms of a and o
h= (square root of) (opposite² + adjacent²)
If A and B vectors are in the same direction, their vector sum _
has magnitude equal to the sum of of the magnitudes of A and B
A vector represents an _______ number of lines. Why?
infinite. A vector is simply a magnitude with a direction meaning it can be picked up and dropped anywhere on the graph and still be represented.
Another name for the tail and head of a vector are _____ and ______
initial point, terminal point
What is the magnitude of the position vector?
is a distance measured in meters in the SI system of units
How the average velocity is obtained by using vector?
is obtained by dividing a vector (displacement) by a scalar (the time interval)
To say a vector is in Rⁿ means?
it has n components in the domain of all real numbers
the displacement of an object is a vector that is drawn from_
its initial position to its final position at some later time
instantaneous velocity is defined as the _
limit of the average velocity as the time interval approaches zero
what are the SI units of the average velocity?
m/s²
A vector quantity is one described by both a_
magnitude (size) and a direction (often an angle relative to one of the coordinate axes)
If vectors A and B point in opposite directions, their vector sum has_
magnitude equal to the difference in the magnitudes of A and B
How is the angle between TWO vectors found?
manipulate this equation by isolating theta to solve for the angle.
the negative of a vector is a vector_ (include how to symbolized)
of the same magnitude but the opposite direction symbolized as -v(arrow top)
sine
opposite/ hypotenuse
tangent
opposite/adjacent
solve sine for opposite
opposite= h sin (delta)
vectors can be added in any_
order
vector
physical quantity that has both magnitude and direction
scalars
quantities that have only magnitude
what symbol is used to represent a position vector?
r (with an arrow above it)
the definition of average acceleration is the_ (include formula)
ratio of the change in velocity over the time interval aavg= (triangle) v/ (traingle) t aavg= (vf-vi)/(tf-ti) *vt, vi, and v have arrow on them)
the sum of two vectors is called_
resultant vector
How determine the final position of an object?
rf=ri + (traingle)r the displacement is the vector that must be added to the position of the object at the beginning of the time interval to give the position of the object at the end of the time interval
what are 3 examples of scalar quantities?
speed mass temperature
vector subtraction can be seen as the_
the addition of a negative vector
the subtraction a vector can be through_
the addition of the opposite of the vector
What represents the direction of a vector?
the arrow points in the direction of the vector quantity
How the direction of the instantaneous velocity compared with the direction of the displacement vector?
the direction of the instantaneous velocity is the same as the direction of the ratio in the limit which is the same as the direction of the displacement vector in the numerator of the ratio
what is the displacement vector of an object?
the displacement vector is a straight-line connection from the object's initial to his final position, not necessarily his actual path
what represents the magnitude in a vector?
the length of the arrow
What is the "tail" of an arrow?
the opposite end to the arrow in a vector
what is the origin in the position vector?
the origin is an arbitrary point in space that has been chosen to represent zero
what is the "head" of an arrow?
the pointed end of the arrow
what is the position of an object represented by a vector?
the position of an object is a vector whose tail is located at the origin and whose head is located at the location of the object
the average velocity of an object is defined as_(include formula)
the ratio of the displacement of the object and the corresponding time interval Vavg=(traingle)r/(traingle) t *both Vavg and r have arrow on them*
component
the scalar number given to the component vectors telling us how big is the component vector and its direction (positive or negative)
what are 3 examples of vector quantities?
velocity acceleration force