The comparison of a quantity with a standard of the same physical quantity.
Physical quantities that have only magnitude and no direction, such as mass, density, and time.
Physical quantities that have both magnitude and direction, such as displacement, force, and velocity.
A set of physical quantities that are completely independent of each other, such as mass and time.
Quantities that can be expressed in terms of fundamental quantities, such as speed and pressure.
The chosen reference standard of measurement in multiples of which a physical quantity is expressed.
Units must be well defined, easily available and reproducible, invariable, and accepted by all.
A system where length, mass, and time are taken as fundamental quantities with base units foot (ft), pound (lb), and second (s).
Length, mass, and time.
All quantities that can be measured, such as time, length, mass, force, and work done.
6.67 × 10⁻¹¹ N·m²/kg².
cos(–60°) = cos(60°) = 1/2.
6.67 × 10⁻⁸ cm³/g·s².
The resultant vector R is maximum, and |R| = |a| + |b|.
sin(300°) = sin(270° + 30°) = –cos(30°) = -√3/2.
P² + Q² - 2PQcos(q) = 113.
2(P² + Q²) = 338.
Sets (A), (B), and (D).
By using the quadratic formula, the solutions are x = 3/2 and x = -4.
The principle of homogeneity of dimensions.
When the angle q = π (180°) between vectors a and b, |R| = |a| - |b|.
Rationalised MKS system.
Using the principle of homogeneity of dimensions.
Kelvin (K).
Units that can be expressed in terms of base units.
î along x-axis, ĵ along y-axis, and k̂ along z-axis.
Joule (J).
The side opposite the right angle, denoted as OP in the triangle OPM.
log(m^n) = n * log(m).
Area = 2πrl (where r = radius and l = length).
3/5.
length × breadth × height = abt.
cos q ≈ 1, tan q ≈ sin q/q.
1 + nx.
Seven base units.
Magnitude = numeric value (n) × unit (u).
The result is a vector parallel to a with modulus |ma| = |m| * |a|.
When they have equal magnitude and are in the same direction, representing the same physical quantity.
s = 2πr.
ML²/T².
That the order of addition does not affect the resultant: A + B = B + A.
L²/T²K.
log(mn) = log(m) + log(n).
Surface Area = 4πr^2 (where r = radius).
The magnitude is maximum for case II (30°).
(side)³.
Dimensionless constants.
It is a measure of change in direction, defined by the ratio of arc length to the length of the line.
cosec q = OP/MP = 5/3.
It states that if vectors are drawn in head to tail fashion, the resultant is defined by a vector drawn from the tail of the first vector to the head of the last vector.
The angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.
It states that if two vectors are placed such that their tails coincide, the diagonal of the parallelogram formed represents the sum of the two vectors.
Area = (1/2) × r^2 × θ (where r = radius and θ is the angle subtended at the center).
The component is given by A · B̂ = |A| cos(θ).
Powers (or exponents) to which the base quantities are raised to represent that quantity.
Force = mass × length × (time)⁻².
(1 - x)ⁿ ≈ 1 - nx.
4/3πr³, where r = radius.
πr²l, where r = radius and l = length.
1/3πr²h, where r = radius and h = height.
Metre (m).
Second (s).
It describes a circle.
Newton (N).
4i + 3j + 5k (25).
Pascal (Pa).
The angle θ between the vectors is given by cos(θ) = (A · B) / (|A||B|).
[F] = MLT⁻².
[M¹ L⁰ T⁰].
|R| = √(a² + b²) and tan(α) = b/a.
A vector of magnitude 1, used to describe direction in space.
Vectors that have the same direction, regardless of their location in space.
1 newton = 10⁵ dyne.
It extends the triangle law to define the addition of more than two vectors.
Yes, they can be in different planes, but if all vectors are coplanar, their resultant must also be in the same plane.
M/L³.
x will be 6.
sin(θ) = perpendicular (PM) / hypotenuse (OP).
|A + B| = √(|A|² + |B|² + 2|A||B|cos(θ)), where θ is the angle between the vectors.
sin²(θ) + cos²(θ) = 1.
D = ad - bc.
Area = πab (where a and b are semi-major and semi-minor axes respectively).
A · B = Ax * Bx + Ay * By + Az * Bz.
ax² + bx + c = 0.
How and which of the base quantities are included in that quantity.
1 - nx.
[M⁰ L⁰ T¹].
A vector has both magnitude and direction.
T = 2π√(R³/GM).
It cannot derive formulas depending on more than three physical quantities or those involving exponential, trigonometric, and logarithmic functions.
Newton (N).
The smaller angle formed when the initial or terminal points of the two vectors are brought together, ranging from 0º to 180º.
Vector shifting is allowed without changing their direction.
T⁻¹.
If p/a = q/b, then (p + q)/(a + b) = (p - q)/(a - b).
log(m/n) = log(m) - log(n).
The dot product is the product of the magnitude of one vector and the magnitude of the component of another vector in the direction of the former vector.
2πr(r + l), where l = length.
πr² + πrl, where l = slant height.
tan q = MP/OM = 3/4.
Metre, kilogram, second.
cos(120°) = cos(180° – 60°) = –cos(60°) = –1/2.
P² + Q² + 2PQcos(q) = 225.
It states that if two vectors are represented as two sides of a triangle, their resultant is represented by the third side.
Vector quantities are usually denoted by boldface letters.
A vector is represented by a directed straight line, indicating its magnitude and direction.
Watt (W).
1 + tan²(θ) = sec²(θ).
Area = (side)^2.
Area = base × height.
The resultant has a magnitude of 113 units.
x = (-b ± √(b² - 4ac)) / (2a).
cos q = OM/OP = 4/5.
Centimetre (cm), gram (g), and second (s).
Because many physical quantities have the same dimensions.
By the ratio of the arc length (s) to the length of the line (r): θ = s/r.
cot q = OM/MP = 4/3.
π rad = 180°.
I, II, III, and IV quadrants.
tan(θ) = perpendicular (PM) / base (OM).
1 + cot²(θ) = cosec²(θ).
Area = (1/2) × (distance between parallel sides) × (sum of parallel sides).
The dot product of a vector by itself, denoted as A · A.
6(side)².
(1 + x)ⁿ ≈ 1 + nx.
tan(210°) = tan(180° + 30°) = tan(30°) = 1/3.
[M⁰ L¹ T⁻¹].
To convert a physical quantity from one system of units to another.
Whether it is a scalar or a vector.
Radians (rad) and degrees.
M.
The magnitude of a vector is a positive scalar and is always positive.
The angle is maximum when the resultant is tangent to the circle.
That the grouping of vectors does not affect the resultant: A + (B + C) = (A + B) + C.
e^x = 1 + x + (x^2/2!) + (x^3/3!) + ...
Area = (1/2) × base × height.
The dot product is zero: A · B = 0.
sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3.
sec q = OP/OM = 5/4.
A vector can be represented as a sum of two or three vectors along predetermined axes.
Vectors that are in opposite directions.
32 ft/s².
A vector that has equal magnitude but is in the opposite direction.
A vector of zero magnitude, resulting from the addition of a vector and its negative vector.
Watt (W).
e^(-x) = 1 - x + (x^2/2!) - (x^3/3!) + ...
Area = πr^2 (where r = radius).
The dot product of unit vectors î, ĵ, k̂ is 1 if they are the same and 0 if they are different.
A vector opposite in direction but equal in magnitude to another vector.
As the addition of a negative vector.
cos(θ) = base (OM) / hypotenuse (OP).
Area = length × breadth.
Yes, the dot product is commutative: A · B = B · A.
