4.3 Vectors

• Vectors are commonly expressed in column form.
• In column notation, the top value represents the horizontal component.
• What is the result of \begin{bmatrix} 2 \\ 5 \end{bmatrix} + \begin{bmatrix} 3 \\ - 1 \end{bmatrix}</latex>?

  $\begin{bmatrix} 5 \\ 4 \end{bmatrix}$
• When multiplying a vector by a scalar, you multiply each component of the vector by the scalar.
• To multiply a vector by a scalar, you multiply each component of the vector by the scalar.


  True
• Match the component of a vector with its scalar multiplication:

  Horizontal component ↔️ $ka$

  Vertical component ↔️ $kb$
• Vectors provide more comprehensive information about a physical quantity than scalars
• In column notation, a vector is represented as $\begin{bmatrix} a \\ b \end{bmatrix}$, where $a$ is the horizontal component and $b$ is the vertical component.
• The formula for vector subtraction in column notation is \begin{bmatrix} a_{1} - a_{2} \\ b_{1} - b_{2} \end{bmatrix}</latex>
• The formula for adding two vectors in column notation is \begin{bmatrix} a_{1} \\ b_{1} \end{bmatrix} + \begin{bmatrix} a_{2} \\ b_{2} \end{bmatrix} = \begin{bmatrix} a_{1} + a_{2} \\ b_{1} + b_{2} \end{bmatrix}</latex>
• If $v =$$\begin{bmatrix} a \\ b \end{bmatrix}$ and $k$ is a scalar, what is the formula for scalar multiplication?


  $k \begin{bmatrix} a \\ b \end{bmatrix} =$$\begin{bmatrix} ka \\ kb \end{bmatrix}$
• What is the definition of parallel vectors?

  Vectors with same or opposite direction
• What theorem is used to calculate the magnitude of a vector?

  Pythagoras' Theorem
• The magnitude of a vector represents its length or size.

  True
• What do position vectors indicate in geometric problems?

  Location of a point
• What two properties does a vector have?

  Magnitude and direction
• What is the magnitude of the vector $\begin{bmatrix} 3 \\ 4 \end{bmatrix}$?


  5
• Describe the movement indicated by the vector \begin{bmatrix} 3 \\ - 2 \end{bmatrix}</latex>.

  Right and down
• To subtract two vectors, you subtract their corresponding components.
• Multiplying a vector by a scalar changes its magnitude but not its direction.

  False
• The formula for subtracting two vectors \begin{bmatrix} a_{1} \\ b_{1} \end{bmatrix}</latex> and $\begin{bmatrix} a_{2} \\ b_{2} \end{bmatrix}$ is $\begin{bmatrix} a_{1} - a_{2} \\ b_{1} - b_{2} \end{bmatrix}$
• Multiplying a vector by a scalar $k$ scales its magnitude by a factor of $k$
• What are the two key properties of a vector?

  Magnitude and direction
• Match the type of quantity with its properties:

  Vector ↔️ Magnitude and direction

  Scalar ↔️ Magnitude only
• What is the formula for vector addition in column notation?

  \begin{bmatrix} a_{1} +$a_{2} \\ b_{1} +$ b_{2} \end{bmatrix}
• What is the formula for subtracting two vectors using their components in column notation?

  $\begin{bmatrix} a_{1} \\ b_{1} \end{bmatrix} - \begin{bmatrix} a_{2} \\ b_{2} \end{bmatrix} =$$\begin{bmatrix} a_{1} - a_{2} \\ b_{1} - b_{2} \end{bmatrix}$
• To multiply a vector by a scalar, you multiply each component of the vector by the scalar.


  True
• If $v =$$\begin{bmatrix} 3 \\ 2 \end{bmatrix}$ and $k =$$4$, then $< latex > 4 \begin{bmatrix} 3 \\ 2 \end{bmatrix} =$$\begin{bmatrix} 12 \\ 8 \end{bmatrix}$</latex>


  True
• The vectors $\begin{bmatrix} 2 \\ 3 \end{bmatrix}$ and $\begin{bmatrix} 4 \\ 6 \end{bmatrix}$ are parallel because $2 \times \begin{bmatrix} 2 \\ 3 \end{bmatrix} =$$\begin{bmatrix} 4 \\ 6 \end{bmatrix}$.


  True
• What is the magnitude of the vector $v =$$\begin{bmatrix} 3 \\ 4 \end{bmatrix}$?


  5
• Match the vector type with its formula:

  Position Vector ↔️ $\vec{OP} =$$\begin{bmatrix} a \\ b \end{bmatrix}$

  Displacement Vector ↔️ $\vec{PQ} =$$\vec{OQ} - \vec{OP}$
• To prove that the midpoints of the sides of a quadrilateral form a parallelogram, one must demonstrate that the displacement vectors between the midpoints are equal and parallel.


  True
• A scalar has both magnitude and direction.

  False
• The horizontal component of a vector can indicate movement left or right.

  True
• To add two vectors, you add their corresponding components.


  True
• What is the result of $\begin{bmatrix} 4 \\ 2 \end{bmatrix} - \begin{bmatrix} 1 \\ 3 \end{bmatrix}$?


  $\begin{bmatrix} 3 \\ - 1 \end{bmatrix}$
• What is the formula for adding two vectors $\begin{bmatrix} a_{1} \\ b_{1} \end{bmatrix}$ and $\begin{bmatrix} a_{2} \\ b_{2} \end{bmatrix}$?


  \begin{bmatrix} a_{1} +$a_{2} \\ b_{1} +$ b_{2} \end{bmatrix}
• What is the formula for multiplying a vector $\begin{bmatrix} a \\ b \end{bmatrix}$ by a scalar $k$?


  $\begin{bmatrix} ka \\ kb \end{bmatrix}$
• If $v =$$\begin{bmatrix} 3 \\ 2 \end{bmatrix}$ and $k =$$4$, then 4v = \begin{bmatrix} 12 \\ 8 \end{bmatrix}</latex>.


  True
• The vector \begin{bmatrix} 3 \\ 4 \end{bmatrix}</latex> has a magnitude of 5 and points northeast.


  True