Vectors - Algebra Of Vectors

Vectors - Algebra of Vectors

Scalars vs Vectors
Representation of Vectors
Addition of Vectors
Subtraction of Vectors
Multiplication of a Vector with a Scalar

Scalars vs Vectors

Scalars: magnitude only
Vectors: magnitude and direction
Example: Distance vs Displacement

Representation of Vectors

Notation: $\vec{a}$
Magnitude: $|\vec{a}|$
Direction: Angle or Unit Vector Example:
Given vector: $\vec{a}=3\hat{i}+4\hat{j}$
Magnitude: $|\vec{a}|=\sqrt{(3^2+4^2)}$
Direction: Angle = $\arctan\left(\frac{4}{3}\right)$

Addition of Vectors

Commutative property: $\vec{a}+\vec{b}=\vec{b}+\vec{a}$
Triangle Law: $\vec{a}+\vec{b}=\vec{c}$
Parallelogram Law: $\vec{a}+\vec{b}=\vec{d}$ Example:
$\vec{a}=3\hat{i}+4\hat{j}$ and $\vec{b}=2\hat{i}-2\hat{j}$
Triangle Law: $\vec{a}+\vec{b}=3\hat{i}+4\hat{j}+2\hat{i}-2\hat{j}=(3+2)\hat{i}+(4-2)\hat{j}$
Parallelogram Law: $\vec{a}+\vec{b}=3\hat{i}+4\hat{j}+(-2\hat{i})+(-2\hat{j})$

Subtraction of Vectors

$\vec{a}-\vec{b}=\vec{a}+(-\vec{b})$ Example:
$\vec{a}=3\hat{i}+4\hat{j}$ and $\vec{b}=2\hat{i}-2\hat{j}$
$\vec{a}-\vec{b}=3\hat{i}+4\hat{j}-(2\hat{i}-2\hat{j})$

Multiplication of a Vector with a Scalar

Scalar multiplication: $c\vec{a} = \vec{a}c$ Example:
$\vec{a}=3\hat{i}+4\hat{j}$
$2\vec{a} = 2(3\hat{i}+4\hat{j})$

Dot Product of Vectors

Geometric interpretation
Formula: $\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta$ Example:
$\vec{a}=3\hat{i}$ and $\vec{b}=4\hat{j}$
$\vec{a} \cdot \vec{b} = (3\hat{i}) \cdot (4\hat{j})$

Cross Product of Vectors

Geometric interpretation
Formula: $\vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin \theta$ Example:
$\vec{a}=3\hat{i}$ and $\vec{b}=4\hat{j}$
$\vec{a} \times \vec{b} = (3\hat{i}) \times (4\hat{j})$

Scalar Triple Product

Formula: $\vec{a} \cdot (\vec{b} \times \vec{c})$
Geometrical interpretation Example:
$\vec{a}=3\hat{i}$, $\vec{b}=4\hat{j}$, $\vec{c}=5\hat{k}$
$\vec{a} \cdot (\vec{b} \times \vec{c})$

Geometric Interpretation of Dot Product

The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them.
The dot product measures the extent to which two vectors are aligned or parallel to each other.
If the dot product is positive, the vectors are pointing in the same general direction.
If the dot product is negative, the vectors are pointing in the opposite general direction.
If the dot product is zero, the vectors are orthogonal or perpendicular to each other. Example:
$\vec{a} = 3\hat{i}+4\hat{j}$ and $\vec{b} = 2\hat{i}-2\hat{j}$
$\vec{a} \cdot \vec{b} = (3\hat{i}+4\hat{j}) \cdot (2\hat{i}-2\hat{j})$

Properties of Dot Product

Distributive property: $(\vec{a} + \vec{b}) \cdot \vec{c} = \vec{a} \cdot \vec{c} + \vec{b} \cdot \vec{c}$
Commutative property: $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$
Multiplicative property: $(c\vec{a}) \cdot \vec{b} = \vec{a} \cdot (c\vec{b}) = c(\vec{a} \cdot \vec{b})$ Example:
$\vec{a} = 3\hat{i}+4\hat{j}$, $\vec{b} = 2\hat{i}-2\hat{j}$, and $\vec{c} = 5\hat{i}-3\hat{j}$
$(\vec{a} + \vec{b}) \cdot \vec{c} = (3\hat{i}+4\hat{j}+2\hat{i}-2\hat{j}) \cdot (5\hat{i}-3\hat{j})$

Geometric Interpretation of Cross Product

The cross product of two vectors is a vector that is perpendicular to both of the original vectors.
The magnitude of the cross product is equal to the product of the magnitudes of the two vectors and the sine of the angle between them.
The direction of the cross product follows the right-hand rule.
The cross product can be used to find areas of parallelograms and volumes of parallelepipeds. Example:
$\vec{a} = 3\hat{i}$ and $\vec{b} = 4\hat{j}$
$\vec{a} \times \vec{b} = (3\hat{i}) \times (4\hat{j})$

Properties of Cross Product

Distributive property: $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$
Anticommutative property: $\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}$
Multiplicative property: $(c\vec{a}) \times \vec{b} = \vec{a} \times (c\vec{b}) = c(\vec{a} \times \vec{b})$ Example:
$\vec{a} = 3\hat{i}+4\hat{j}$, $\vec{b} = 2\hat{i}-2\hat{j}$, and $\vec{c} = 5\hat{i}-3\hat{j}$
$\vec{a} \times (\vec{b} + \vec{c}) = (3\hat{i}+4\hat{j}) \times (2\hat{i}-2\hat{j}+5\hat{i}-3\hat{j})$

Scalar Triple Product

The scalar triple product of three vectors is a scalar quantity.
It can be calculated by taking the dot product of one vector with the cross product of the other two vectors.
It is used to find the volume of a parallelepiped formed by the three vectors. Example:
$\vec{a} = 3\hat{i}$, $\vec{b} = 4\hat{j}$, and $\vec{c} = 5\hat{k}$
$\vec{a} \cdot (\vec{b} \times \vec{c})$

Properties of Scalar Triple Product

Anticommutative property: $\vec{a} \cdot (\vec{b} \times \vec{c}) = -\vec{b} \cdot (\vec{a} \times \vec{c}) = \vec{c} \cdot (\vec{a} \times \vec{b})$
Distributive property: $(\vec{a} + \vec{b}) \cdot (\vec{c} \times \vec{d}) = \vec{a} \cdot (\vec{c} \times \vec{d}) + \vec{b} \cdot (\vec{c} \times \vec{d})$ Example:
$\vec{a} = 3\hat{i}+4\hat{j}$, $\vec{b} = 2\hat{i}-2\hat{j}$, $\vec{c} = 5\hat{i}-3\hat{j}$, and $\vec{d} = -\hat{k}$
$(\vec{a} + \vec{b}) \cdot (\vec{c} \times \vec{d})$

Magnitude of a Vector

The magnitude or length of a vector is a scalar value that represents the size of the vector.
It can be calculated using the Pythagorean theorem for two-dimensional vectors or the distance formula for three-dimensional vectors. Example:
$\vec{a} = 3\hat{i}+4\hat{j}$
$|\vec{a}| = \sqrt{3^2 + 4^2}$

Unit Vector

A unit vector is a vector that has a magnitude of 1.
It is often used to represent directions or orientations.
It is denoted by placing a hat (^) above the vector symbol. Example:
$\vec{a} = 3\hat{i}+4\hat{j}$
Unit vector in the direction of $\vec{a}$: $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$

Projection of a Vector

The projection of a vector $\vec{a}$ onto a vector $\vec{b}$ is a vector that represents the component of $\vec{a}$ that is in the direction of $\vec{b}$.
It can be calculated using the dot product and the magnitude of $\vec{b}$. Example:
$\vec{a} = 3\hat{i}+4\hat{j}$ and $\vec{b} = 2\hat{i}-2\hat{j}$
Projection of $\vec{a}$ onto $\vec{b}$: $\text{proj}_\vec{b}(\vec{a}) = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b}$

Component of a Vector

The component of a vector $\vec{a}$ along a vector $\vec{b}$ is a scalar that represents the magnitude of the projection of $\vec{a}$ onto $\vec{b}$.
It can be calculated using the dot product and the magnitude of $\vec{b}$. Example:
$\vec{a} = 3\hat{i}+4\hat{j}$ and $\vec{b} = 2\hat{i}-2\hat{j}$
Component of $\vec{a}$ along $\vec{b}$: $\text{comp}_\vec{b}(\vec{a}) = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$

Magnitude of a Vector

The magnitude or length of a vector is a scalar value that represents the size of the vector.
It can be calculated using the Pythagorean theorem for two-dimensional vectors or the distance formula for three-dimensional vectors.
Magnitude of a vector $\vec{a}$ in two dimensions: $|\vec{a}| = \sqrt{a_x^2 + a_y^2}$
Magnitude of a vector $\vec{a}$ in three dimensions: $|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}$ Example:
$\vec{a} = 3\hat{i}+4\hat{j}$
$|\vec{a}| = \sqrt{3^2 + 4^2}$

Unit Vector

A unit vector is a vector that has a magnitude of 1.
It is often used to represent directions or orientations.
It is denoted by placing a hat (^) above the vector symbol.
Unit vector in the direction of $\vec{a}$: $\hat{a} = \frac{\vec{a}}{|\vec{a}|}$ Example:
$\vec{a} = 3\hat{i}+4\hat{j}$
Unit vector in the direction of $\vec{a}$: $\hat{a} = \frac{(3\hat{i}+4\hat{j})}{\sqrt{3^2 + 4^2}}$

Projection of a Vector

The projection of a vector $\vec{a}$ onto a vector $\vec{b}$ is a vector that represents the component of $\vec{a}$ that is in the direction of $\vec{b}$.
It can be calculated using the dot product and the magnitude of $\vec{b}$.
Projection of $\vec{a}$ onto $\vec{b}$: $\text{proj}_\vec{b}(\vec{a}) = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b}$ Example:
$\vec{a} = 3\hat{i}+4\hat{j}$ and $\vec{b} = 2\hat{i}-2\hat{j}$
Projection of $\vec{a}$ onto $\vec{b}$: $\text{proj}_\vec{b}(\vec{a}) = \left(\frac{(3\hat{i}+4\hat{j}) \cdot (2\hat{i}-2\hat{j})}{|2\hat{i}-2\hat{j}|^2}\right)(2\hat{i}-2\hat{j})$

Component of a Vector

The component of a vector $\vec{a}$ along a vector $\vec{b}$ is a scalar that represents the magnitude of the projection of $\vec{a}$ onto $\vec{b}$.
It can be calculated using the dot product and the magnitude of $\vec{b}$.
Component of $\vec{a}$ along $\vec{b}$: $\text{comp}_\vec{b}(\vec{a}) = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$ Example:
$\vec{a} = 3\hat{i}+4\hat{j}$ and $\vec{b} = 2\hat{i}-2\hat{j}$
Component of $\vec{a}$ along $\vec{b}$: $\text{comp}_\vec{b}(\vec{a}) = \frac{(3\hat{i}+4\hat{j}) \cdot (2\hat{i}-2\hat{j})}{|2\hat{i}-2\hat{j}|}$

Vector Equations

A vector equation is an equation in which vectors are involved.
It represents geometric properties such as lines, planes or other higher dimensional objects.
Example: $\vec{r} = \vec{a} + t\vec{b}$ represents a line in vector form, where $\vec{r}$ is a position vector, $\vec{a}$ is a known vector, $\vec{b}$ is a direction vector, and $t$ is a parameter representing the position on the line.

Scalar Triple Product

The scalar triple product of three vectors is a scalar quantity.
It can be calculated by taking the dot product of one vector with the cross product of the other two vectors.
It is used to find the volume of a parallelepiped formed by the three vectors.
Scalar triple product of $\vec{a}$, $\vec{b}$, and $\vec{c}$: $(\vec{a} \times \vec{b}) \cdot \vec{c}$ Example:
$\vec{a} = 3\hat{i}$, $\vec{b} = 4\hat{j}$, and $\vec{c} = 5\hat{k}$
$(\vec{a} \times \vec{b}) \cdot \vec{c} = ((3\hat{i}) \times (4\hat{j}})) \cdot (5\hat{k})$

Properties of Scalar Triple Product

Anticommutative property: $\vec{a} \cdot (\vec{b} \times \vec{c}) = -\vec{b} \cdot (\vec{a} \times \vec{c}) = \vec{c} \cdot (\vec{a} \times \vec{b})$
Distributive property: $(\vec{a} + \vec{b}) \cdot (\vec{c} \times \vec{d}) = \vec{a} \cdot (\vec{c} \times \vec{d}) + \vec{b} \cdot (\vec{c} \times \vec{d})$ Example:
$\vec{a} = 3\hat{i}+4\hat{j}$, $\vec{b} = 2\hat{i}-2\hat{j}$, $\vec{c} = 5\hat{i}-3\hat{j}$, and $\