3.2 Vector Equations

    Cards (53)

    • What two properties define a vector?
      Magnitude and direction
    • A vector is represented geometrically as an arrow.
    • In column vector notation, a vector AB\overrightarrow{AB} from A(x1,y1)A(x_{1}, y_{1}) to B(x2,y2)B(x_{2}, y_{2}) is expressed as AB=\overrightarrow{AB} =(x2x1y2y1) \begin{pmatrix} x_{2} - x_{1} \\ y_{2} - y_{1} \end{pmatrix}, where x2x1x_{2} - x_{1} and y2y1y_{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?
      r=\overrightarrow{r} =a+ \overrightarrow{a} +td t\overrightarrow{d}
    • Match the terms in the vector equation of a line with their meanings:
      r\overrightarrow{r} ↔️ Position vector of any point on the line
      a\overrightarrow{a} ↔️ Position vector of a specific point on the line
      d\overrightarrow{d} ↔️ Direction vector of the line
      tt ↔️ 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)P(x, y, z)?

      OP=\overrightarrow{OP} =(xyz) \begin{pmatrix} x \\ y \\ z \end{pmatrix}
    • What do the components of the vector (34)\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?
      r=\overrightarrow{r} =a+ \overrightarrow{a} +td t\overrightarrow{d}
    • In the vector equation of a line, r\overrightarrow{r} represents the position vector of any point on the line
    • What does the direction vector d\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:
      r\overrightarrow{r} ↔️ Position vector of any point on the line
      a\overrightarrow{a} ↔️ Position vector of a specific point on the line
      d\overrightarrow{d} ↔️ Direction vector of the line
      tt ↔️ Scalar parameter
    • What is the vector equation of a line passing through (1,2)(1, 2) with direction (31)\begin{pmatrix} 3 \\ - 1 \end{pmatrix}?

      r=\overrightarrow{r} =(12)+ \begin{pmatrix} 1 \\ 2 \end{pmatrix} +t(31) 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?
      n(ra)=\overrightarrow{n} \cdot (\overrightarrow{r} - \overrightarrow{a}) =0 0
    • In the vector equation of a plane, n\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 (211)\begin{pmatrix} 2 \\ 1 \\ - 1 \end{pmatrix} and a point A(1,2,3)A(1, 2, 3)?

      (211)((xyz)(123))=\begin{pmatrix} 2 \\ 1 \\ - 1 \end{pmatrix} \cdot (\begin{pmatrix} x \\ y \\ z \end{pmatrix} - \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}) =0 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 tt 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 AB\overrightarrow{AB} for points A(1,2)A(1, 2) and B(4,6)B(4, 6)?

      AB=\overrightarrow{AB} =(34) \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 r=\overrightarrow{r} =a+ \overrightarrow{a} +td t\overrightarrow{d}.
    • What does a\overrightarrow{a} represent in the vector equation of a line?

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

      Normal vector to the plane
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