Vectors

Vectors are quantities that have both magnitude and direction.
They are often represented by an arrow where the length of the arrow represents the magnitude and the direction of the arrow represents the direction of the vector.
Vectors can be added, subtracted, and multiplied by scalars.
Scalar multiplication involves multiplying the magnitude of the vector by a scalar.
Vectors can also be multiplied using the dot product and cross product. Examples:
Displacement of an object
Force exerted on an object
Velocity of an object Equations:
Magnitude of a vector: |𝑎⃗ | = √(𝑎𝑥² + 𝑎𝑦²)
Unit vector: 𝑢⃗ = 𝑎⃗ / |𝑎⃗ |

Vector Addition

Vectors can be added together using the parallelogram law.
To add two vectors, place the tail of the second vector at the head of the first vector.
The sum of the two vectors is the vector that extends from the tail of the first vector to the head of the second vector.
The order in which the vectors are added does not matter; the result will be the same.
Vector addition is both commutative and associative. Example:
A vector 𝑎⃗ = 3𝑖̂ + 2𝑗̂ and 𝑏⃗ = -𝑖̂ + 4𝑗̂. Find the sum 𝑐⃗ = 𝑎⃗ + 𝑏⃗. Equation:
𝑐⃗ = 𝑎⃗ + 𝑏⃗ = (3𝑖̂ + 2𝑗̂) + (-𝑖̂ + 4𝑗̂)

Vector Subtraction

Vector subtraction is similar to vector addition, but in the opposite direction.
To subtract two vectors, we add the negative of the second vector to the first vector.
The result is a vector that points from the tail of the second vector to the head of the first vector. Example:
A vector 𝑎⃗ = 3𝑖̂ + 2𝑗̂ and 𝑏⃗ = -𝑖̂ + 4𝑗̂. Find the difference 𝑑⃗ = 𝑎⃗ - 𝑏⃗. Equation:
𝑑⃗ = 𝑎⃗ - 𝑏⃗ = (3𝑖̂ + 2𝑗̂) - (-𝑖̂ + 4𝑗̂)

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number).
To multiply a vector by a scalar, multiply each component of the vector by the scalar.
The direction of the resulting vector remains the same, but the magnitude is scaled by the scalar. Example:
Multiply the vector 𝑎⃗ = 2𝑖̂ + 3𝑗̂ by the scalar 5. Equation:
𝑏⃗ = 5(2𝑖̂ + 3𝑗̂)

Dot Product

The dot product, also known as the scalar product, is a binary operation that takes two vectors and produces a scalar.
The dot product of two vectors 𝑎⃗ and 𝑏⃗ is denoted as 𝑎⃗ · 𝑏⃗.
The dot product is computed by multiplying the corresponding components of the vectors and summing the products.
The dot product is commutative: 𝑎⃗ · 𝑏⃗ = 𝑏⃗ · 𝑎⃗.
Geometrically, the dot product can be interpreted as the product of the magnitudes of the two vectors and the cosine of the angle between them. Example:
Calculate the dot product of 𝑎⃗ = 2𝑖̂ + 3𝑗̂ and 𝑏⃗ = -𝑖̂ + 4𝑗̂. Equation:
𝑎⃗ · 𝑏⃗ = (2𝑖̂ + 3𝑗̂) · (-𝑖̂ + 4𝑗̂)

Properties of the Dot Product

Distributive Property: 𝑎⃗ · (𝑏⃗ + 𝑐⃗) = 𝑎⃗ · 𝑏⃗ + 𝑎⃗ · 𝑐⃗
Scalar Multiplication: (𝑎𝑏⃗) · 𝑐⃗ = 𝑎⃗ · (𝑏𝑐⃗) = 𝑎𝑏𝑐⃗ ·
Dot Product with the Zero Vector: 𝑎⃗ · 𝑣⃗ = 0 Example:
Verify the distributive property: (𝑎⃗ · 𝑏⃗)⃗ + (𝑎⃗ · 𝑐⃗)⃗ = 𝑎⃗ · (𝑏⃗ + 𝑐⃗)⃗. Equation:
(𝑎⃗ · 𝑏⃗)⃗ + (𝑎⃗ · 𝑐⃗)⃗ = 𝑎⃗ · (𝑏⃗ + 𝑐⃗)⃗ = ?

Cross Product

The cross product, also known as the vector product, is a binary operation that takes two vectors and produces another vector.
The cross product of two vectors 𝑎⃗ and 𝑏⃗ is denoted as 𝑎⃗ × 𝑏⃗.
The cross product is computed using the determinant of a 3x3 matrix involving the components of the vectors.
The resulting vector is orthogonal to both the original vectors and its direction is determined by the right-hand rule. Example:
Calculate the cross product of 𝑎⃗ = 2𝑖̂ + 3𝑗̂ + 𝑘̂ and 𝑏⃗ = -𝑖̂ + 4𝑗̂ - 2𝑘̂. Equation:
𝑎⃗ × 𝑏⃗ = (2𝑖̂ + 3𝑗̂ + 𝑘̂) × (-𝑖̂ + 4𝑗̂ - 2𝑘̂)

Properties of the Cross Product

The cross product is distributive: 𝑎⃗ × (𝑏⃗ + 𝑐⃗) = 𝑎⃗ × 𝑏⃗ + 𝑎⃗ × 𝑐⃗
Scalar Multiplication: 𝑎⃗ × (𝑏𝑐⃗) = (𝑎⃗ × 𝑏⃗)𝑐⃗ = 𝑎𝑏⃗ × 𝑐⃗
The cross product of two parallel vectors is the zero vector: 𝑎⃗ × 𝑏⃗ = 𝑣⃗ Example:
Verify the distributive property: 𝑎⃗ × (𝑏⃗ + 𝑐⃗) = 𝑎⃗ × 𝑏⃗ + 𝑎⃗ × 𝑐⃗. Equation:
𝑎⃗ × (𝑏⃗ + 𝑐⃗) = 𝑎⃗ × 𝑏⃗ + 𝑎⃗ × 𝑐⃗ = ?

Applications of Vectors: Displacement

Vectors are commonly used to represent the displacement of an object.
Displacement is a vector quantity that represents the change in position of an object from its initial position to its final position.
Displacement is defined as the vector that points from the initial position to the final position of the object.
Displacement can be measured in terms of magnitude and direction or using components in a coordinate system. Example:
An object moves 5 meters east and then 3 meters north. What is the displacement of the object? Equation:
𝑎⃗ = 5𝑖̂ + 3𝑗̂

Applications of Vectors: Force

Vectors are used to represent forces acting on an object.
Force is a vector quantity that represents the physical interaction between two objects.
A force is characterized by its magnitude and direction.
Forces can be added and subtracted using vector addition and subtraction. Example:
Two forces, 𝐹⃗1 = 5𝑖̂ - 2𝑗̂ and 𝐹⃗2 = -3𝑖̂ + 𝑗̂, act on an object. What is the resulting force? Equation:
𝐹⃗ = 𝐹⃗1 + 𝐹⃗2

Applications of Vectors: Velocity

Vectors are commonly used to represent the velocity of an object.
Velocity is a vector quantity that represents the rate of change of displacement with respect to time.
Velocity is defined as the vector that points in the direction of motion and has a magnitude equal to the speed of the object.
Velocity can be measured in terms of magnitude and direction or using components in a coordinate system. Example:
An object moves 10 meters east in 2 seconds. What is the velocity of the object? Equation:
𝑣⃗ = 𝑑⃗ / 𝑡

</section>
<section style="font-size: 50px";>
<h2 id="vtp-identity">VTP Identity</h2>
<ul>
<li>The VTP Identity is a mathematical identity involving vectors and their cross product.</li>
<li>It states that the cross product of the cross product of two vectors is equal to the vector multiplied by the dot product of the two vectors minus the dot product of the vector with the first vector.
VTP Identity Equation:</li>
<li>𝑎⃗ × (𝑏⃗ × 𝑐⃗) = (𝑎⃗ · 𝑐⃗)𝑏⃗ - (𝑎⃗ · 𝑏⃗)𝑐⃗
Example:</li>
<li>Given 𝑎⃗ = 2𝑖̂ + 3𝑗̂ + 𝑘̂, 𝑏⃗ = -𝑖̂ + 4𝑗̂ - 2𝑘̂, and 𝑐⃗ = 𝑖̂ + 𝑘̂.</li>
<li>Calculate the cross product of 𝑎⃗ and (𝑏⃗ × 𝑐⃗) using the VTP Identity.
Equation:</li>
<li>𝑎⃗ × (𝑏⃗ × 𝑐⃗) = (𝑎⃗ · 𝑐⃗)𝑏⃗ - (𝑎⃗ · 𝑏⃗)𝑐⃗ = ?</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="vector-projection">Vector Projection</h2>
<ul>
<li>The vector projection is a method used to find the component of a vector in the direction of another vector.</li>
<li>It allows us to break down a vector into two components: the parallel component and the perpendicular component.</li>
<li>The parallel component represents the part of the vector that lies in the same direction as the other vector.</li>
<li>The perpendicular component represents the part of the vector that is perpendicular to the other vector.
Example:</li>
<li>Given 𝑎⃗ = 5𝑖̂ + 3𝑗̂ and 𝑏⃗ = 2𝑖̂ - 𝑗̂.</li>
<li>Find the vector projection of 𝑎⃗ onto 𝑏⃗.
Equation:</li>
<li>𝑎⃗𝑝⃗𝑟⃗𝑜⃗𝑗⃗𝑒⃗𝑐⃗𝑡⃗𝑖⃗𝑜⃗𝑛⃗ = ((𝑎⃗ · 𝑏⃗) / |𝑏⃗ |²)𝑏⃗ = ?</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="angle-between-vectors">Angle Between Vectors</h2>
<ul>
<li>The angle between two vectors can be calculated using the dot product.</li>
<li>The dot product of two vectors can be written as the product of their magnitudes and the cosine of the angle between them.</li>
<li>Using this equation, we can solve for the angle between the vectors.
Angle Equation:</li>
<li>cos(θ) = 𝑎⃗ · 𝑏⃗ / (|𝑎⃗ |⋅|𝑏⃗ |)
Example:</li>
<li>Given 𝑎⃗ = 2𝑖̂ + 3𝑗̂ and 𝑏⃗ = -𝑖̂ + 4𝑗̂.</li>
<li>Find the angle between 𝑎⃗ and 𝑏⃗.
Equation:</li>
<li>cos(θ) = 𝑎⃗ · 𝑏⃗ / (|𝑎⃗ |⋅|𝑏⃗ |) = ?</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="scalar-triple-product">Scalar Triple Product</h2>
<ul>
<li>The scalar triple product is a mathematical operation involving three vectors.</li>
<li>It is calculated by taking the dot product of one vector with the cross product of the other two vectors.</li>
<li>The scalar triple product can be used to determine whether three vectors are coplanar or not.</li>
<li>If the scalar triple product is equal to zero, the three vectors are coplanar; otherwise, they are not coplanar.
Scalar Triple Product Equation:</li>
<li>(𝑎⃗ × 𝑏⃗) · 𝑐⃗
Example:</li>
<li>Given 𝑎⃗ = 2𝑖̂ + 3𝑗̂ + 𝑘̂, 𝑏⃗ = -𝑖̂ + 4𝑗̂ - 2𝑘̂, and 𝑐⃗ = 𝑖̂ + 𝑘̂.</li>
<li>Calculate the scalar triple product: (𝑎⃗ × 𝑏⃗) · 𝑐⃗.
Equation:</li>
<li>(𝑎⃗ × 𝑏⃗) · 𝑐⃗ = ?</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="applications-of-vectors-work">Applications of Vectors: Work</h2>
<ul>
<li>Vectors are used to represent work, which is the product of the force applied on an object and the displacement of the object.</li>
<li>Work is a scalar quantity, but it can be represented using vectors.</li>
<li>The work done on an object is given by the dot product of the force vector and the displacement vector.</li>
<li>Positive work is done when the force and displacement are in the same direction, while negative work is done when they are in opposite directions.
Work Equation:</li>
<li>𝑊 = 𝐹⃗ · 𝑑⃗</li>
<li>Where 𝑊 is the work, 𝐹⃗ is the force vector, and 𝑑⃗ is the displacement vector.
Example:</li>
<li>A force of 10 N is applied on an object that moves a distance of 5 meters in the same direction as the force.</li>
<li>Find the work done on the object.
Equation:</li>
<li>𝑊 = 𝐹⃗ · 𝑑⃗ = ?</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="applications-of-vectors-torque">Applications of Vectors: Torque</h2>
<ul>
<li>Vectors are used to represent torque, which is the rotational equivalent of force.</li>
<li>Torque is the product of the force applied perpendicular to the axis of rotation and the distance from the axis of rotation.</li>
<li>Torque is a vector quantity that has both magnitude and direction.</li>
<li>The direction of the torque vector is given by the right-hand rule, where the thumb points in the direction of the torque vector when the fingers curl in the direction of rotational motion.
Torque Equation:</li>
<li>𝜏 = 𝐹⃗ × 𝑟⃗</li>
<li>Where 𝜏 is the torque, 𝐹⃗ is the force vector, and 𝑟⃗ is the position vector or distance from the axis of rotation.
Example:</li>
<li>A force of 10 N is applied perpendicular to a lever arm with a length of 2 meters.</li>
<li>Find the torque generated by the force.
Equation:</li>
<li>𝜏 = 𝐹⃗ × 𝑟⃗ = ?</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="parametric-equations">Parametric Equations</h2>
<ul>
<li>Parametric equations are a way of representing curves using equations that describe the coordinates of points in terms of parameters.</li>
<li>The parameters are typically denoted as 𝑡 or 𝑠 and represent the position of the point along the curve.</li>
<li>Parametric equations are used to represent functions that are not easily expressed in terms of a single variable.</li>
<li>They are often used to represent curves in 2D or 3D space.
Example:</li>
<li>The parametric equations for a circle of radius 𝑟 centered at the origin are 𝑥 = 𝑟cos(𝑡) and 𝑦 = 𝑟sin(𝑡), where 𝑡 is the parameter.</li>
<li>Plot the circle using these parametric equations.
Equation:</li>
<li>𝑥 = 𝑟cos(𝑡)</li>
<li>𝑦 = 𝑟sin(𝑡)</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="applications-of-vectors-motion-in-a-plane">Applications of Vectors: Motion in a Plane</h2>
<ul>
<li>Vectors are used to represent motion in a plane, where objects move in both the x and y directions.</li>
<li>Velocity and acceleration can</li>
</ul>
</section>

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