4.3 Using vectors to describe motion of an object

• A scalar has both magnitude and direction

  False
• The direction of a vector is the angle it makes with the positive x-axis

  True
• What does a position vector describe?

  The location of an object
• An object with a position vector $\mathbf{r} =$$(3t, 4t)$ has a velocity vector $\mathbf{v} =$$(3, 4)$ m/s, indicating its constant speed and direction.
• What does the direction of a vector represent?

  The angle with the x-axis
• The displacement vector describes the change in position of an object


  True
• To calculate the displacement, subtract the corresponding components
• A position vector locates an object in a coordinate system
• $\mathbf{r}_{f} =$$(x_{f}, y_{f})$ represents the final position
• What is the formula to express displacement as a vector difference?

  \mathbf{d} = \mathbf{r}_{f} - \mathbf{r}_{i}</latex>
• An object moving from (2, 3) to (5, 7) has a displacement of (3, 4).


  True
• If a vector $\mathbf{v} =$$(3, 4)$, what is its magnitude?


  5 units/second
• What is the formula for adding two motion vectors \mathbf{v}_{1} = (x_{1}, y_{1})</latex> and $\mathbf{v}_{2} =$$(x_{2}, y_{2})$?


  $\mathbf{v} =$$(x_{1} +$$x_{2}, y_{1} +$$y_{2})$
• Adding $\mathbf{v}_{1} =$$(3, 4)$ and $\mathbf{v}_{2} =$$( - 1, 2)$ results in $\mathbf{v} =$$(2, 6)$.


  True
• A ball thrown at 10 m/s at 30 degrees has a horizontal component of 8.66 m/s.

  True
• Steps to solve real-world motion problems with vectors:

  1️⃣ Sketch the problem and label vector components

  2️⃣ Apply relevant formulas

  3️⃣ Calculate unknowns

  4️⃣ Write a concise summary
• What is a vector in the context of motion description?

  A quantity with magnitude and direction
• How is the magnitude of a vector $\mathbf{v}$ calculated?


  |\mathbf{v}| = \sqrt{v_{x}^{2} + v_{y}^{2}}</latex>
• An object moving 10 meters east at a 30-degree angle can be represented as a vector with magnitude 10 and direction 30 degrees.
• What is the formula for the velocity vector $\mathbf{v}$?


  \mathbf{v} = \frac{d\mathbf{r}}{dt}</latex>
• The magnitude of a vector is always positive


  True
• Displacement is the vector that describes the change in position
• Match the quantity type with its characteristics:

  Scalar ↔️ Magnitude only

  Vector ↔️ Magnitude and direction
• What is the formula for displacement as a vector difference?

  $\mathbf{d} =$$\mathbf{r}_{f} - \mathbf{r}_{i}$
• An object moving from (2, 3) to (5, 7) has a displacement of (3, 4).


  True
• Match the concept with its definition:

  Displacement Vector ↔️ Change in position

  Final Position ↔️ Endpoint of displacement

  Initial Position ↔️ Starting point of displacement
• The direction of a vector is measured as the angle from the positive x-axis
• The direction of a vector $\mathbf{v} =$$(3, 4)$ is approximately 53.13 degrees.


  True
• The magnitude of a vector is defined as its length
• After sketching the problem, the next step is to apply relevant formulas
• The magnitude of a vector is its length
• What is the formula for calculating the direction $\theta$ of a vector?


  $\theta =$$\tan^{ - 1}\left(\frac{v_{y}}{v_{x}}\right)$
• The velocity vector is the rate of change of the position vector with respect to time


  True
• How does a vector differ from a scalar?

  A vector has direction
• How is displacement calculated as a vector difference?

  $\mathbf{d} =$$\mathbf{r}_{f} - \mathbf{r}_{i}$
• The magnitude of a vector $\mathbf{v} =$$(v_{x}, v_{y})$ is calculated as \sqrt{v_{x}^{2} + v_{y}^{2}}
• Displacement can be expressed as the vector difference between the final and initial positions
• What is the definition of the displacement vector?

  Change in position
• $\mathbf{r}_{f} =$$(x_{f}, y_{f})$ represents the final position
• What does the magnitude of a vector represent?

  Length of the vector