4.3 Using vectors to describe motion of an object

• Vectors are characterized by both magnitude and direction
• Vectors can be notated using angle brackets, such as $\langle x, y \rangle$
• The magnitude of a vector $\langle x, y \rangle$ is calculated using the Pythagorean theorem
• Vector addition involves adding corresponding components of vectors
• Scalar multiplication scales a vector by multiplying each component
• Vectors are used to represent quantities like velocity, displacement, and force
• Position, velocity, and acceleration can be represented as vectors
• Position vectors indicate an object's location relative to an origin
• Acceleration vectors define the rate of change of velocity
• Vectors are used to analyze and predict the motion of objects accurately
• The magnitude of a vector $\langle x, y \rangle$ is calculated using the Pythagorean theorem
• Vector addition involves adding corresponding components of vectors
• Scalar multiplication scales a vector by multiplying each component
• The magnitude of a vector \langle x, y \rangle</latex> is calculated using the Pythagorean theorem
• Vector addition involves adding corresponding components of vectors
• Scalar multiplication scales a vector by multiplying each component
• Position, velocity, and acceleration can be represented as vectors
• Position vectors indicate an object's location relative to an origin
• Acceleration vectors define the rate of change of velocity
• Vectors are used to analyze and predict the motion of objects accurately
• What three physical quantities can be represented as vectors in two or three dimensions?

  Position, velocity, acceleration
• Position vectors indicate an object's location relative to an origin
• What does a velocity vector describe?

  Speed and direction
• Acceleration vectors define the rate of change of velocity
• Vectors allow for the accurate analysis of motion by considering both speed and direction.
• Match the vector with its description:

  Position ↔️ Location relative to an origin

  Velocity ↔️ Speed and direction of movement

  Acceleration ↔️ Rate of change of velocity
• What are the two primary vector operations discussed in the study material?

  Vector addition and scalar multiplication
• Vector addition involves adding the corresponding components
• Scalar multiplication scales the magnitude of a vector by a scalar value.
• What is the result of adding $\langle 1, 2 \rangle +$$\langle 3, 4 \rangle$?


  $\langle 4, 6 \rangle$
• Multiplying the vector $\langle 1, 2 \rangle$ by the scalar $2$ results in \langle 2, 4 \rangle</latex>
• Steps to resolve a vector into its components:

  1️⃣ Identify the magnitude and angle of the vector

  2️⃣ Calculate the x-component using $v_{x} =$$|\vec{v}| \cos(\theta)$

  3️⃣ Calculate the y-component using $v_{y} =$$|\vec{v}| \sin(\theta)$
• What are the x and y components of a vector with magnitude $10$ and angle $30^\circ$?


  $\langle 8.66, 5 \rangle$
• When resolving a vector, the angle $\theta$ is measured with respect to the positive x-axis.
• What is the defining characteristic of projectile motion?

  Influence of gravity
• In projectile motion, the horizontal component of velocity remains constant
• The acceleration vector in projectile motion is $\vec{a} =$$\langle 0, - g \rangle$.
• If an object is launched at $v_{0} =$$20 m / s$ at $\theta =$$45^\circ$, what is $v_{x}$?


  $14.14 m / s$
• The vertical displacement in projectile motion is given by the equation $y(t) =$$v_{y}(0) t - \frac{1}{2} g t^{2}$, where $g$ is the gravitational constant
• What remains constant in projectile motion over time?

  $v_{x}$