3.2 Vector Equations

• What two properties define a vector?

  Magnitude and direction
• A vector is represented geometrically as an arrow.
• In column vector notation, a vector $\overrightarrow{AB}$ from $A(x_{1}, y_{1})$ to $B(x_{2}, y_{2})$ is expressed as $\overrightarrow{AB} =$$\begin{pmatrix} x_{2} - x_{1} \\ y_{2} - y_{1} \end{pmatrix}$, where $x_{2} - x_{1}$ and $y_{2} - y_{1}$ are its components
• What do the components of a vector represent in column vector notation?

  Movement along axes
• A vector is a quantity with both magnitude and direction
• Vectors build upon coordinate geometry by giving direction and magnitude to coordinate differences.
• What is a position vector?

  Origin to a point
• A direction vector indicates the direction of a vector without specifying its location
• Position and direction vectors are essential for describing points and directions in vector geometry.
• What is the vector equation of a line?

  $\overrightarrow{r} =$$\overrightarrow{a} +$$t\overrightarrow{d}$
• Match the terms in the vector equation of a line with their meanings:

  $\overrightarrow{r}$ ↔️ Position vector of any point on the line

  $\overrightarrow{a}$ ↔️ Position vector of a specific point on the line

  $\overrightarrow{d}$ ↔️ Direction vector of the line

  $t$ ↔️ Scalar parameter
• In the vector equation of a line, the direction vector indicates the direction
• Vectors extend coordinate geometry by adding direction and magnitude to coordinate differences.
• What is the position vector of a point $P(x, y, z)$?


  $\overrightarrow{OP} =$$\begin{pmatrix} x \\ y \\ z \end{pmatrix}$
• What do the components of the vector $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ indicate in terms of movement?


  3 units along the x-axis and 4 units along the y-axis
• Vectors build upon coordinate geometry by giving direction and magnitude
• A position vector points from the origin to a specific point in space.
• What does a direction vector indicate about a vector?

  Direction
• Understanding direction vectors helps describe the orientation of lines and planes.
• What is the vector equation of a line?

  $\overrightarrow{r} =$$\overrightarrow{a} +$$t\overrightarrow{d}$
• In the vector equation of a line, $\overrightarrow{r}$ represents the position vector of any point on the line
• What does the direction vector $\overrightarrow{d}$ indicate in the vector equation of a line?


  The direction of the line
• Match the component of the vector equation of a line with its description:

  $\overrightarrow{r}$ ↔️ Position vector of any point on the line

  $\overrightarrow{a}$ ↔️ Position vector of a specific point on the line

  $\overrightarrow{d}$ ↔️ Direction vector of the line

  $t$ ↔️ Scalar parameter
• What is the vector equation of a line passing through $(1, 2)$ with direction $\begin{pmatrix} 3 \\ - 1 \end{pmatrix}$?


  $\overrightarrow{r} =$$\begin{pmatrix} 1 \\ 2 \end{pmatrix} +$$t\begin{pmatrix} 3 \\ - 1 \end{pmatrix}$
• The vector equation of a plane uses a normal vector and a position vector.
• What is the vector equation of a plane?

  $\overrightarrow{n} \cdot (\overrightarrow{r} - \overrightarrow{a}) =$$0$
• In the vector equation of a plane, $\overrightarrow{n}$ represents the normal vector to the plane
• The normal vector to a plane is perpendicular to the plane.
• What is the vector equation of a plane with normal vector $\begin{pmatrix} 2 \\ 1 \\ - 1 \end{pmatrix}$ and a point $A(1, 2, 3)$?


  $\begin{pmatrix} 2 \\ 1 \\ - 1 \end{pmatrix} \cdot (\begin{pmatrix} x \\ y \\ z \end{pmatrix} - \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}) =$$0$
• The simplified vector equation of a plane in the example is 2x + y - z = 1</latex>.
• How do you determine if a point lies on a line using its vector equation?

  Solve for $t$ and check consistency
• A point lies on a plane if substituting its coordinates into the plane's vector equation satisfies the equation.
• What two properties define a vector?

  Magnitude and direction
• Match the term with its description:

  Magnitude ↔️ Length of the vector

  Direction ↔️ Orientation of the vector

  Column vector notation ↔️ Representation of a vector using components
• What is the vector $\overrightarrow{AB}$ for points $A(1, 2)$ and $B(4, 6)$?


  $\overrightarrow{AB} =$$\begin{pmatrix} 3 \\ 4 \end{pmatrix}$
• What does a position vector define about a point in space?

  Location
• The vector equation of a line is $\overrightarrow{r} =$$\overrightarrow{a} +$$t\overrightarrow{d}$.
• What does $\overrightarrow{a}$ represent in the vector equation of a line?


  Position vector of a specific point
• The vector equation of a plane is $\overrightarrow{n} \cdot (\overrightarrow{r} - \overrightarrow{a}) =$$0$.
• What does $\overrightarrow{n}$ represent in the vector equation of a plane?


  Normal vector to the plane