3.1 Vector Algebra

• Vectors are represented using boldface letters or an arrow over the letter
• Vectors in two dimensions have three components

  False
• What is the formula for adding two vectors $\mathbf{a}$ and $\mathbf{b}$?


  $\mathbf{a} +$$\mathbf{b} =$ \begin{pmatrix} a_{1} +$b_{1} \\ a_{2} +$$b_{2} \\ a_{3} +$ b_{3} \end{pmatrix}
• Vector addition is performed by adding corresponding components
• Vector subtraction involves subtracting corresponding components
• What is the formula for scalar multiplication of a vector $\mathbf{a}$ by a scalar $k$?


  k\mathbf{a} = \begin{pmatrix} ka_{1} \\ ka_{2} \\ ka_{3} \end{pmatrix}</latex>
• Vector addition is commutative, meaning \mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}
• Vector addition is not commutative

  False
• What are the three basic vector operations?

  Addition, subtraction, scalar multiplication
• Steps to perform vector addition

  1️⃣ Ensure the vectors have the same dimension

  2️⃣ Add the corresponding components together
• For vectors \mathbf{a} = \begin{pmatrix} a_{1} \\ a_{2} \end{pmatrix}</latex> and $\mathbf{b} =$$\begin{pmatrix} b_{1} \\ b_{2} \end{pmatrix}$, their sum is \begin{pmatrix} a_{1} + b_{1} \\ a_{2} + b_{2} \end{pmatrix}
• The sum of $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ - 1 \end{pmatrix}$ is $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$
• What is the formula for adding two vectors $\mathbf{a}$ and $\mathbf{b}$?


  $\mathbf{a} +$$\mathbf{b} =$ \begin{pmatrix} a_{1} +$b_{1} \\ a_{2} +$ b_{2} \end{pmatrix}
• Vector addition is commutative.
• To perform vector addition, ensure the vectors have the same dimension
• Steps to perform vector addition

  1️⃣ Ensure the vectors have the same dimension.

  2️⃣ Add the corresponding components together.
• What does scalar multiplication change in a vector?

  Magnitude
• Scalar multiplication reverses the direction of a vector if the scalar is negative.
• The magnitude of a vector is scaled by the scalar's absolute value
• What is the formula for subtracting two vectors $\mathbf{a}$ and $\mathbf{b}$?


  \mathbf{a} - \mathbf{b} = \begin{pmatrix} a_{1} - b_{1} \\ a_{2} - b_{2} \end{pmatrix}</latex>
• Subtracting a vector $\mathbf{b}$ from $\mathbf{a}$ is equivalent to adding $- \mathbf{b}$ to $\mathbf{a}$.
• What does a boldface letter represent in vector notation?

  A vector without components
• Match the vector notation with its description:

  Boldface Letter ↔️ Represents a vector without components

  Arrow Notation ↔️ Another way to denote a vector

  Component Notation ↔️ Lists components within brackets
• What is the commutative property of vector addition?

  \mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}</latex>
• What is the formula for adding two vectors $\mathbf{a} =$$\begin{pmatrix} a_{1} \\ a_{2} \end{pmatrix}$ and $\mathbf{b} =$$\begin{pmatrix} b_{1} \\ b_{2} \end{pmatrix}$?


  $\mathbf{a} +$$\mathbf{b} =$ \begin{pmatrix} a_{1} +$b_{1} \\ a_{2} +$ b_{2} \end{pmatrix}
• Vector addition is a commutative operation.
• To perform vector addition, we add corresponding components
• What is the result of adding \mathbf{a} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}</latex> and $\mathbf{b} =$$\begin{pmatrix} 1 \\ - 1 \end{pmatrix}$?


  $\mathbf{a} +$$\mathbf{b} =$$\begin{pmatrix} 3 \\ 2 \end{pmatrix}$
• Adding two vectors involves summing their corresponding entries.
• Match the vector operation with its formula:

  Vector Addition ↔️ $\mathbf{a} +$$\mathbf{b}$

  Vector Subtraction ↔️ $\mathbf{a} - \mathbf{b}$
• What is the result of subtracting $\mathbf{a} =$$\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\mathbf{b} =$$\begin{pmatrix} 1 \\ - 1 \end{pmatrix}$?


  $\mathbf{a} - \mathbf{b} =$$\begin{pmatrix} 1 \\ 4 \end{pmatrix}$
• Vector addition is commutative, meaning the order of addition does not affect the result.
• What is the formula for vector addition?

  $\mathbf{a} +$$\mathbf{b} =$ \begin{pmatrix} a_{1} +$b_{1} \\ a_{2} +$ b_{2} \end{pmatrix}
• What does it mean for vector addition to be commutative?

  $\mathbf{a} +$$\mathbf{b} =$$\mathbf{b} +$$\mathbf{a}$
• What is the formula for vector addition again?

  $\mathbf{a} +$$\mathbf{b} =$ \begin{pmatrix} a_{1} +$b_{1} \\ a_{2} +$ b_{2} \end{pmatrix}
• What are the distributive properties of scalar multiplication?

  k(\mathbf{a} + \mathbf{b}) = k\mathbf{a} + k\mathbf{b}</latex> and $(k + l)\mathbf{a} =$$k\mathbf{a} +$$l\mathbf{a}$
• In scalar multiplication, the direction of the vector reverses if the scalar is negative.
• What is the formula for vector subtraction?

  $\mathbf{a} - \mathbf{b} =$$\begin{pmatrix} a_{1} - b_{1} \\ a_{2} - b_{2} \end{pmatrix}$
• The magnitude of a vector is its length or distance from the origin to the vector's endpoint.
• What is the formula for the magnitude of a 2D vector?

  |\mathbf{v}| = \sqrt{v_{1}^{2} + v_{2}^{2}}</latex>