Units and Measurements

Length SI

meter (m)

Mass SI

kilogram (kg)

Time SI

Second (s)

Temperature SI

Kelvin (K)

Current SI

Ampere (A)

Amount of substance SI

Mole (mol)

Luminous Intensity SI

Candela (cd)

kilo (k)

10³

centi (c)

10^-2

milli (m)

10^-3

micro (μ)

10^-6

nano (n)

10^-9

mega (M)

10^6

giga (G)

10^9

Length dimension

L

Mass dimension

M

Time dimension

T

Density dimension

ML^-3

Area dimension

L²

Volume dimension

L³

Speed dimension

LT^-1

Force dimension

MLT^-2

Sig fig rules

1. final answer can’t have more sig figs than least precise value

2. multiplication and division: result has the sme number of sig figs as least precise input

3. addition and subtraction: result have the no more decimal places than the least precise measurement

Net displacement is found by…

adding each movement/vector together

If vectors aren’t aligned with the x or y plane, what formulas can we use?

x = dcos(theta)

y=dsin(theta)

How do you find the displacement of a vector?

You find the displacement of a vector by adding its x and y components, which can be calculated using x = dcos(theta) and y = dsin(theta). Then, use the pythagorean theorem.

How do you calculate absolute and relative uncertainty?

Absolute uncertainty is the measure of the doubt in a measurement, typically given as a fixed quantity, while relative uncertainty is the ratio of absolute uncertainty to the measurement itself, often expressed as a percentage.

What is a vector? What is a scalar?

A vector is a quantity that has both magnitude and direction, while a scalar is a quantity that has only magnitude without direction, such as temperature or mass.

What is a dot product?

The dot product is an algebraic operation between two vectors that yields a scalar, calculated as the product of their magnitudes and the cosine of the angle between them.

What is a cross product?

The cross product is a mathematical operation on two vectors in three-dimensional space that results in another vector perpendicular to both, with a magnitude equal to the area of the parallelogram formed by the two vectors.

What does orthogonal mean?

Orthogonal refers to two vectors being perpendicular to each other, meaning their dot product equals zero and they have no component in the same direction.

How do you find the Y component of a vector?

The Y component of a vector is found by multiplying the vector's magnitude by the sine of the angle it makes with the horizontal axis.

How do you find the X component of a vector?

To find the X component of a vector, you multiply the vector's magnitude by the cosine of the angle it makes with the horizontal axis.

How do you find the angle of a vector?

To find the angle of a vector, you use the inverse tangent function (arctan) of the ratio of the Y component to the X component of the vector.

What is a unit vector?

A unit vector is a vector with a magnitude of one, used to indicate direction. It is often denoted by a caret (^) and can be obtained by dividing a vector by its magnitude.

What are equivalent vectors? paralell? antiparalell?

Equivalent vectors are vectors that have the same magnitude and direction, meaning they point in the same direction and cover the same distance regardless of their position. Parallel vectors point in the same or opposite directions and have the same line of action, while antiparallel vectors point in opposite directions but are parallel.

What are some general properties of scalars?

Scalars are quantities that have only magnitude and no direction, such as temperature, mass, and speed. They can be added, subtracted, multiplied, or divided using standard arithmetic operations.

What are some general properties of vectors?

Vectors have magnitude and direction, can be added and subtracted using the head-to-tail method, and obey the commutative and associative properties. Additionally, they can be represented as components in a coordinate system.

Briefly decribe how you perform vector addition

Vector addition involves placing the tail of one vector at the head of another, forming a polygon. The resultant vector is drawn from the tail of the first vector to the head of the last vector.

Briefly decribe how you perform vector multiplication

Vector multiplication involves either the dot product or the cross product. The dot product results in a scalar and measures the extent to which two vectors align, while the cross product results in a vector that is perpendicular to the plane formed by the original vectors.

Briefly decribe how you perform vector subtraction

Vector subtraction is performed by adding the opposite of the vector to be subtracted. This involves reversing the direction of the vector and then following the head-to-tail method for addition.

How do you find the magnitude of a vector?

To find the magnitude of a vector, you use the Pythagorean theorem, which involves taking the square root of the sum of the squares of its components.

How do you find the value of a resultant?

To find the value (magnitude) of a resultant vector, follow these steps:

1. Calculate the X and Y components for each individual vector using the formulas x = d \cos(\theta) and y = d \sin(\theta), where d is the vector's magnitude and \theta is its angle with the horizontal axis.

2. Sum all the X-components to get the total X-component of the resultant vector (R_x).

3. Sum all the Y-components to get the total Y-component of the resultant vector (R_y).

4. Use the Pythagorean theorem to find the magnitude of the resultant vector (R) by taking the square root of the sum of the squares of its total X and Y components: R = \sqrt{Rx^2 + Ry^2}.

What are the different components of a vector?

A vector has two main components: the X-component, which represents its horizontal direction, and the Y-component, which represents its vertical direction. These components can be determined using trigonometric functions based on the vector's magnitude and angle.

What are the different cooridnate systems that can represent a vector?

Vectors can be represented using Cartesian coordinates, polar coordinates, and cylindrical or spherical coordinate systems, each providing a different method for expressing the vector's position and direction.

How do you represent a vector graphically?

A vector can be represented graphically as an arrow, where the length of the arrow indicates its magnitude and the direction of the arrow shows its direction. The tail of the arrow represents the starting point, while the head indicates the endpoint of the vector.

How do you represent a vector in writing?

A vector can be represented in writing by describing its components, magnitude, and direction, often using boldface letters or arrows to indicate the vector quantity. Alternatively, it may be expressed using coordinates in a specific system, such as (X, Y) in Cartesian coordinates.

How do you perform head to tail addition?

Head to tail addition involves placing the tail of one vector at the head of another vector and drawing a new vector from the tail of the first vector to the head of the last vector, representing the resultant vector.

How do you perform the paralellogram method?

The parallelogram method involves placing two vectors such that they originate from the same point, forming a parallelogram. The resultant vector is then drawn from the shared starting point to the opposite corner of the parallelogram.