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Physics

Vectors

Multiplication of a Vector by a Number

11
⚡ Quick Summary
Multiplying a vector by a number changes its magnitude. If the number is positive, the direction stays the same. If it's negative, the direction flips!
<b>b</b> = k<b>a</b>
  • If vector a has magnitude a, and k is a number, then vector b = ka has magnitude |ka|.
  • If k > 0, the direction of b is the same as a.
  • If k < 0, the direction of b is opposite to a.
  • Multiplying by -1 just reverses the vector's direction. Vectors a and -a have the same magnitude but opposite directions.

Subtraction of Vectors

11
⚡ Quick Summary
Subtracting a vector is like adding its opposite! Just flip the direction of the vector you're subtracting and then add them together.
<b>a</b> - <b>b</b> = <b>a</b> + (-<b>b</b>)
  • a - b is defined as a + (-b).
  • To subtract b from a, reverse the direction of b and then add it to a.

Resolution of Vectors

11
⚡ Quick Summary
Breaking down a vector into its parts along the X, Y, and Z axes. It's like figuring out how much the vector 'points' in each direction.
<b>a</b> = a cos(α) <b>i</b> + a cos(β) <b>j</b> + a cos(γ) <b>k</b>
  • A vector a in the X-Y plane can be resolved into components along the X and Y axes.
  • The projection of a on the X-axis is a cos(alpha), where alpha is the angle between a and the X-axis.
  • The projection of a on the Y-axis is a cos(beta), where beta is the angle between a and the Y-axis.
  • If i and j are unit vectors along the X and Y axes, then a = a cos(alpha) i + a cos(beta) j.
  • In 3D space, a = a cos(alpha) i + a cos(beta) j + a cos(gamma) k, where alpha, beta, and gamma are the angles made by a with the X, Y, and Z axes respectively, and k is the unit vector along the Z-axis.
  • a cos(alpha), a cos(beta), and a cos(gamma) are the components of a along the X, Y, and Z axes respectively.

Cross Product or Vector Product of Two Vectors

Class 11
⚡ Quick Summary
The cross product of two vectors results in another vector that's perpendicular to both original vectors. Think of it like finding a direction that's 'out' of the plane formed by the two vectors.
|a × b| = ab sin(θ)
  • The cross product of two vectors `a` and `b` is written as `a × b`.
  • The magnitude of the cross product is given by |`a × b`| = ab sin(θ), where a and b are the magnitudes of vectors `a` and `b` respectively, and θ is the smaller angle between them.
  • The direction of `a × b` is perpendicular to the plane containing `a` and `b`. The direction is determined by the right-hand thumb rule.
  • Right-hand thumb rule: Point your fingers along vector `a` and curl them towards vector `b`. Your thumb points in the direction of `a × b`.
  • The cross product is anti-commutative: `a × b = -b × a`. Switching the order of the vectors reverses the direction of the resulting vector.
  • The cross product follows the distributive law: `a × (b + c) = a × b + a × c`.
  • The cross product does *not* follow the associative law: `a × (b × c) ≠ (a × b) × c`.

Cross Product of Unit Vectors

Class 11
⚡ Quick Summary
The cross products of the unit vectors i, j, and k follow a cyclic pattern. `i x j = k`, `j x k = i`, `k x i = j`. If you reverse the order, you get the negative: `j x i = -k`
i × j = k, j × k = i, k × i = j
  • `i × j = k`
  • `j × k = i`
  • `k × i = j`
  • `i × i = j × j = k × k = 0`
  • `j × i = -k`
  • `k × j = -i`
  • `i × k = -j`

Cross Product in Component Form

Class 11
⚡ Quick Summary
To find the cross product of two vectors when you know their components, you can use the distributive property and the unit vector cross products, or the determinant method. It's a bit messy, but it works!
a × b = (a<sub>y</sub>b<sub>z</sub> - a<sub>z</sub>b<sub>y</sub>)i + (a<sub>z</sub>b<sub>x</sub> - a<sub>x</sub>b<sub>z</sub>)j + (a<sub>x</sub>b<sub>y</sub> - a<sub>y</sub>b<sub>x</sub>)k
  • If `a = axi + ayj + azk` and `b = bxi + byj + bzk`, then `a × b = (aybz - azby)i + (azbx - axbz)j + (axby - aybx)k`.

Right-Handed Coordinate System

Class 11
⚡ Quick Summary
A right-handed coordinate system is the standard way to set up your x, y, and z axes. If you point your right-hand fingers from the x-axis to the y-axis, your thumb will point along the positive z-axis.
i × j = k
  • In a right-handed coordinate system, `i × j = k`, `j × k = i`, and `k × i = j`. This ensures consistency with the right-hand rule.
  • The positive direction of the Z-axis is chosen in such a way that i × j = k.

Magnitude of a Resultant Vector (Two Perpendicular Vectors)

11
⚡ Quick Summary
When you have two vectors that are at right angles to each other (like going straight and then turning perfectly), the overall length (magnitude) of the combined effect (resultant) is found using a square root.
A = √(A_x² + A_y²)
If a vector is the resultant of two perpendicular vectors with magnitudes A_x and A_y, then the magnitude of the resultant vector A is given by the Pythagorean theorem.

Direction of a Resultant Vector (Two Perpendicular Vectors)

11
⚡ Quick Summary
The angle of the combined vector (resultant) can be calculated using trigonometry, specifically the tangent function. It tells you the direction of the combined effect.
tan α = A_y / A_x
The angle α between the resultant vector A and the X-axis is given by the tangent function, relating the y-component to the x-component.

Components of a Vector

11
⚡ Quick Summary
Think of a vector as an arrow. You can break it down into how much it points horizontally (x-component) and how much it points vertically (y-component).
A_x = A cos θ, A_y = A sin θ
The x-component of a vector A is A cos θ, and the y-component is A sin θ, where θ is the angle the vector makes with the x-axis.

Vector Addition (Component Method)

11
⚡ Quick Summary
To add vectors, break each one into its horizontal and vertical parts, then add all the horizontal parts together and all the vertical parts together. The result is the horizontal and vertical parts of the final combined vector.
R_x = A_x + B_x + C_x + ..., R_y = A_y + B_y + C_y + ...
To find the resultant of multiple vectors, resolve each vector into its x and y components. Then, sum all the x-components to get the x-component of the resultant, and sum all the y-components to get the y-component of the resultant.

Vector Addition - Triangle Rule

Class 11
⚡ Quick Summary
The triangle rule helps add vectors by placing them head-to-tail. The resultant vector connects the tail of the first to the head of the last.
R = A + B (vector addition)
If two vectors are represented by two sides of a triangle in magnitude and direction taken in the same order, then the third side represents the resultant vector in magnitude and direction taken in the reverse order. This is especially useful when visualizing the combination of displacements or velocities.

Scalar (Dot) Product

Class 11
⚡ Quick Summary
The dot product multiplies the magnitudes of two vectors and the cosine of the angle between them. It results in a scalar value.
A · B = |A| |B| cos θ
The scalar or dot product of two vectors is defined as the product of the magnitude of the two vectors and the cosine of the angle between them. It's useful for finding the component of one vector along the direction of another.

Vector (Cross) Product

Class 11
⚡ Quick Summary
The cross product multiplies the magnitudes of two vectors and the sine of the angle between them. The result is a vector perpendicular to both original vectors.
|A x B| = |A| |B| sin θ
The vector or cross product of two vectors results in a vector perpendicular to the plane containing the two original vectors. Its magnitude is the product of the magnitudes of the vectors and the sine of the angle between them. The direction is determined by the right-hand rule.

Unit Vectors

Class 11
⚡ Quick Summary
Unit vectors have a magnitude of 1 and point along the coordinate axes. They help express vectors in terms of components.
|i| = |j| = |k| = 1
Unit vectors (i, j, k) point along the x, y, and z axes, respectively. Any vector can be written as a sum of scalar multiples of these unit vectors (e.g. A = Ax i + Ay j + Az k ). This makes vector operations easier to perform.

Component of a Vector

Class 11
⚡ Quick Summary
A vector's component along a line is its 'shadow' on that line.
Ax = A cos θ (x-component), Ay = A sin θ (y-component)
The component of a vector A along a certain direction is the projection of the vector onto that direction. It represents the 'effective' part of the vector acting along that line. Components are scalars and can be positive or negative.