3.1 Vector Algebra

    Cards (53)

    • 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 a\mathbf{a} and b\mathbf{b}?

      a+\mathbf{a} +b= \mathbf{b} = \begin{pmatrix} a_{1} +b1a2+ b_{1} \\ a_{2} +b2a3+ 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 a\mathbf{a} by a scalar kk?

      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 b=\mathbf{b} =(b1b2) \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 (23)\begin{pmatrix} 2 \\ 3 \end{pmatrix} and (11)\begin{pmatrix} 1 \\ - 1 \end{pmatrix} is (32)\begin{pmatrix} 3 \\ 2 \end{pmatrix}
    • What is the formula for adding two vectors a\mathbf{a} and b\mathbf{b}?

      a+\mathbf{a} +b= \mathbf{b} = \begin{pmatrix} a_{1} +b1a2+ 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 a\mathbf{a} and b\mathbf{b}?

      \mathbf{a} - \mathbf{b} = \begin{pmatrix} a_{1} - b_{1} \\ a_{2} - b_{2} \end{pmatrix}</latex>
    • Subtracting a vector b\mathbf{b} from a\mathbf{a} is equivalent to adding b- \mathbf{b} to a\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 a=\mathbf{a} =(a1a2) \begin{pmatrix} a_{1} \\ a_{2} \end{pmatrix} and b=\mathbf{b} =(b1b2) \begin{pmatrix} b_{1} \\ b_{2} \end{pmatrix}?

      a+\mathbf{a} +b= \mathbf{b} = \begin{pmatrix} a_{1} +b1a2+ 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 b=\mathbf{b} =(11) \begin{pmatrix} 1 \\ - 1 \end{pmatrix}?

      a+\mathbf{a} +b= \mathbf{b} =(32) \begin{pmatrix} 3 \\ 2 \end{pmatrix}
    • Adding two vectors involves summing their corresponding entries.
    • Match the vector operation with its formula:
      Vector Addition ↔️ a+\mathbf{a} +b \mathbf{b}
      Vector Subtraction ↔️ ab\mathbf{a} - \mathbf{b}
    • What is the result of subtracting a=\mathbf{a} =(23) \begin{pmatrix} 2 \\ 3 \end{pmatrix} and b=\mathbf{b} =(11) \begin{pmatrix} 1 \\ - 1 \end{pmatrix}?

      ab=\mathbf{a} - \mathbf{b} =(14) \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?
      a+\mathbf{a} +b= \mathbf{b} = \begin{pmatrix} a_{1} +b1a2+ b_{1} \\ a_{2} + b_{2} \end{pmatrix}
    • What does it mean for vector addition to be commutative?
      a+\mathbf{a} +b= \mathbf{b} =b+ \mathbf{b} +a \mathbf{a}
    • What is the formula for vector addition again?
      a+\mathbf{a} +b= \mathbf{b} = \begin{pmatrix} a_{1} +b1a2+ 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)a=(k + l)\mathbf{a} =ka+ k\mathbf{a} +la l\mathbf{a}
    • In scalar multiplication, the direction of the vector reverses if the scalar is negative.
    • What is the formula for vector subtraction?
      ab=\mathbf{a} - \mathbf{b} =(a1b1a2b2) \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>