1.10 Vectors

• What is the formula for scalar multiplication of a vector by a scalar $k$?


  $k\mathbf{a} =$$\begin{pmatrix} ka_{1} \\ ka_{2} \end{pmatrix}$
• Match the vector operation with its formula:

  Addition ↔️ $\mathbf{a} +$$\mathbf{b} =$ \begin{pmatrix} a_{1} +$b_{1} \\ a_{2} +$ b_{2} \end{pmatrix}

  Subtraction ↔️ $\mathbf{a} - \mathbf{b} =$$\begin{pmatrix} a_{1} - b_{1} \\ a_{2} - b_{2} \end{pmatrix}$

  Scalar Multiplication ↔️ $k\mathbf{a} =$$\begin{pmatrix} ka_{1} \\ ka_{2} \end{pmatrix}$
• Given $\mathbf{a} =$$\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ and $\mathbf{b} =$$\begin{pmatrix} 1 \\ 4 \end{pmatrix}$, what is $\mathbf{a} +$$\mathbf{b}$?


  $\begin{pmatrix} 4 \\ 6 \end{pmatrix}$
• The magnitude of a vector $\mathbf{a} =$$\begin{pmatrix} a_{1} \\ a_{2} \end{pmatrix}$ is calculated using the formula $|\mathbf{a}| =$ \sqrt{a_{1}^{2} + a_{2}^{2}}, which gives the length of the vector.
• The direction of a vector $\mathbf{a} =$$\begin{pmatrix} a_{1} \\ a_{2} \end{pmatrix}$ is given by \theta = \arctan\left(\frac{a_{2}}{a_{1}}\right)</latex>.
• Given vector $\mathbf{a} =$$\begin{pmatrix} 3 \\ 4 \end{pmatrix}$, what is its magnitude?


  $5$
• Given vector $\mathbf{a} =$$\begin{pmatrix} 3 \\ 4 \end{pmatrix}$, what is its direction?


  $53.13^\circ$
• A unit vector has a magnitude of 1 and points in the same direction as the original vector.
• What is the formula to find a unit vector $\hat{\mathbf{a}}$ from a vector $\mathbf{a}$?


  $\hat{\mathbf{a}} =$$\frac{\mathbf{a}}{|\mathbf{a}|}$
• Steps to calculate a unit vector

  1️⃣ Find the magnitude of the original vector.

  2️⃣ Divide each component of the original vector by its magnitude.
• Find the unit vector of $\mathbf{a} =$$\begin{pmatrix} 4 \\ 3 \end{pmatrix}$.


  $\begin{pmatrix} 4 / 5 \\ 3 / 5 \end{pmatrix}$
• A unit vector always has a magnitude of 1.
• What does the magnitude-direction form specify about a vector?

  Length and angle
• To convert a vector with magnitude 5 and angle 30° to coordinate form, you use the formulas $x =$$5 \cos(30^\circ)$ and $y =$$5 \sin(30^\circ)$ to find the coordinates.
• What does the vector notation $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$ represent?


  Components along x, y, z axes
• The coordinate form of a vector is denoted as (x, y, z)
• A column vector is denoted as $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$, where x, y, and z are the components along the axes
• The magnitude-direction form specifies the length and direction
• Match the vector form with its notation:

  Column ↔️ $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$

  Coordinate ↔️ $(x, y, z)$

  Magnitude-Direction ↔️ $|v|\angle\theta$
• What is the x-component of a vector with magnitude 5 and direction 30° in coordinate form?

  $5 \cos(30^\circ)$
• Vectors can be added, subtracted, and multiplied by scalars using component-wise operations.
• To add two vectors, you add their corresponding components
• What is the formula for subtracting vector $\mathbf{b}$ from $\mathbf{a}$?


  $\mathbf{a} - \mathbf{b}$
• When multiplying a vector by a scalar, each component is multiplied by the scalar.
• The formula for vector addition is \mathbf{a} + \mathbf{b} = \begin{pmatrix} a_{1} + b_{1} \\ a_{2} + b_{2} \end{pmatrix}</latex>, which sums the corresponding components
• If $\mathbf{a} =$$\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ and $\mathbf{b} =$$\begin{pmatrix} 1 \\ 4 \end{pmatrix}$, what is $\mathbf{a} +$$\mathbf{b}$?


  $\begin{pmatrix} 4 \\ 6 \end{pmatrix}$
• Match the vector operation with its formula:

  Addition ↔️ $\mathbf{a} +$$\mathbf{b} =$ \begin{pmatrix} a_{1} +$b_{1} \\ a_{2} +$ b_{2} \end{pmatrix}

  Subtraction ↔️ $\mathbf{a} - \mathbf{b} =$$\begin{pmatrix} a_{1} - b_{1} \\ a_{2} - b_{2} \end{pmatrix}$

  Scalar Multiplication ↔️ $k\mathbf{a} =$$\begin{pmatrix} ka_{1} \\ ka_{2} \end{pmatrix}$
• A column vector is denoted as $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$, where x, y, and z are the components along the axes.
• In coordinate form, a vector is written as $(x, y, z)$ representing the coordinates in three-dimensional
• How is the direction of a vector specified in magnitude-direction form?

  By an angle
• Order the vector forms based on their notation complexity:

  1️⃣ Column

  2️⃣ Coordinate

  3️⃣ Magnitude-Direction
• What does the notation $|v|\angle\theta$ represent in magnitude-direction form?


  Magnitude and direction
• Vectors can be represented in various forms, including column vectors, coordinate form, and magnitude-direction form.
• A column vector is denoted as $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$, where x, y, and z are the components along the x, y, and z axes.
• How is a vector written in coordinate form?

  $(x, y, z)$
• In magnitude-direction form, the direction is often given as an angle or direction cosines.
• Steps to convert a vector with magnitude 5 and direction 30° to coordinate form

  1️⃣ Calculate x = |v| \cos \theta</latex>

  2️⃣ Calculate $y =$$|v| \sin \theta$

  3️⃣ The vector in coordinate form is $(x, y)$
• What are the three forms in which vectors can be represented?

  Column, coordinate, magnitude-direction
• In a column vector, the components are implicitly defined by the x, y, and z values.
• What is a key characteristic of a magnitude-direction form vector?

  Explicit magnitude and direction