Cards (61)

  • What is a vector in mathematics?
    A quantity with magnitude and direction
  • What does the xx component of a vector represent?

    Movement in the horizontal direction
  • Vectors are equal if their components are equal

    True
  • How do you add two vectors \vec{a} = \begin{pmatrix} x_1 \\ y_1 \end{pmatrix}</latex> and b=\vec{b} =(x2y2) \begin{pmatrix} x_{2} \\ y_{2} \end{pmatrix}?

    Add their corresponding components
  • What is the result of adding a=\vec{a} =(34) \begin{pmatrix} 3 \\ 4 \end{pmatrix} and b=\vec{b} =(12) \begin{pmatrix} 1 \\ - 2 \end{pmatrix}?

    (42)\begin{pmatrix} 4 \\ 2 \end{pmatrix}
  • What is the general form of a vector v\vec{v} as a column matrix?

    v=\vec{v} =(xy) \begin{pmatrix} x \\ y \end{pmatrix}
  • Match the vector operation with its general formula:
    Addition ↔️ a+\vec{a} +b= \vec{b} = \begin{pmatrix} x_{1} +x2y1+ x_{2} \\ y_{1} + y_{2} \end{pmatrix}
    Subtraction ↔️ ab=\vec{a} - \vec{b} =(x1x2y1y2) \begin{pmatrix} x_{1} - x_{2} \\ y_{1} - y_{2} \end{pmatrix}
  • What is the result of multiplying \vec{a} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}</latex> by the scalar k=k =4 4?

    (128)\begin{pmatrix} 12 \\ 8 \end{pmatrix}
  • A vector is a quantity that has both magnitude and direction
  • Vectors are often represented as column matrices
  • A vector \( \vec{v} \) can be expressed as \begin{pmatrix} x \\ y \end{pmatrix}
  • Match the vector component with its description:
    xx ↔️ Horizontal movement
    yy ↔️ Vertical movement
  • A vector has both magnitude and direction
  • What does the yy component of a vector represent?

    Movement in the vertical direction
  • When adding vectors, a+\vec{a} +b= \vec{b} = \begin{pmatrix} x_{1} +x2y1+ x_{2} \\ y_{1} + y_{2} \end{pmatrix}, which adds the corresponding
  • Vectors are equal if and only if their corresponding components are equal.

    True
  • What does scalar multiplication do to a vector's magnitude if the scalar is positive?
    Scales the magnitude
  • The magnitude of a vector represents its length.

    True
  • What does the vector (34)\begin{pmatrix} 3 \\ 4 \end{pmatrix} represent in terms of movement?

    3 units horizontally, 4 units vertically
  • Vectors are represented using components for horizontal and vertical movement.
    True
  • What movement does the vector (34)\begin{pmatrix} 3 \\ 4 \end{pmatrix} represent?

    3 horizontal, 4 vertical
  • A vector \( \vec{v} \) is generally written as \begin{pmatrix} x \\ y \end{pmatrix}
  • What is the general formula for adding two vectors?
    a+\vec{a} +b= \vec{b} = \begin{pmatrix} x_{1} +x2y1+ x_{2} \\ y_{1} + y_{2} \end{pmatrix}
  • What is the formula for scalar multiplication of a vector?
    k\vec{v} = \begin{pmatrix} kx \\ ky \end{pmatrix}</latex>
  • Steps to add or subtract vectors
    1️⃣ Add or subtract corresponding components
    2️⃣ Combine the results into a new vector
  • To multiply a vector v=\vec{v} =(xy) \begin{pmatrix} x \\ y \end{pmatrix} by a scalar \(k\), multiply each component
  • Two vectors are parallel if they have the same magnitude.
    False
  • What is the mathematical condition for two vectors \( \vec{a} \) and \( \vec{b} \) to be parallel?
    \( \vec{a} = k\vec{b} \)
  • What is the additional condition for two vectors to be collinear besides being parallel?
    Share a common starting point
  • Collinear vectors must have the same starting point.

    True
  • To multiply a vector by a scalar, you only multiply one component of the vector.
    False
  • Scalar multiplication changes the direction of a vector if the scalar is positive.
    False
  • The magnitude of a vector is always a positive number.

    True
  • Vectors are collinear if they lie on the same line
  • Two vectors are parallel if they have the same magnitude.
    False
  • Match the vector property with its description:
    Direction of parallel vectors ↔️ Same or opposite
    Position of collinear vectors ↔️ Must lie on the same line
  • The vectors \( \begin{pmatrix} 2 \\ 3 \end{pmatrix} \) and \( \begin{pmatrix} 4 \\ 6 \end{pmatrix} \) are parallel because \( \begin{pmatrix} 4 \\ 6 \end{pmatrix} = 2 \times \begin{pmatrix} 2 \\ 3 \end{pmatrix} \). The scalar \( k \) is 2
  • When multiplying a vector by a scalar \( k \), each component of the vector is multiplied by k
  • What is the relationship between parallel and collinear vectors in terms of their direction?
    Same or opposite
  • Match the vector property with its characteristic:
    Direction of parallel vectors ↔️ Same or opposite
    Position of collinear vectors ↔️ Must lie on the same line