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What does the position vector 𝑟⃗ (𝑡) represent in a three-dimensional Cartesian coordinate system?
What does the position vector 𝑟⃗ (𝑡) represent in a three-dimensional Cartesian coordinate system?
- The change in position over time
- The velocity of an object
- The distance from the origin to a point
- The location of a point relative to the origin (correct)
Which of the following correctly describes displacement?
Which of the following correctly describes displacement?
- It is a change of position from initial to final location. (correct)
- It measures only the shortest distance to the origin.
- It indicates the total distance traveled by an object.
- It is a scalar quantity representing distance.
How can the displacement vector 𝛥𝑟⃗ be expressed using unit-vector notation?
How can the displacement vector 𝛥𝑟⃗ be expressed using unit-vector notation?
- ∆𝑟⃗ = (𝑥2−𝑥1)𝑖̇̂ + (𝑦2−𝑦1)𝑗̇̂ + (𝑧2−𝑧1)𝑘̂ (correct)
- ∆𝑟⃗ = (𝑥2+𝑦2+𝑧2)𝑖̇̂ + (𝑥1+𝑦1+𝑧1)𝑗̇̂ + (𝑥2+𝑦1+𝑧2)𝑘̂
- ∆𝑟⃗ = (𝑥1 + 𝑥2)𝑖̇̂ + (𝑦1 + 𝑦2)𝑗̇̂ + (𝑧1 + 𝑧2)𝑘̂
- ∆𝑟⃗ = (𝑥1−𝑥2)𝑖̇̂ + (𝑦1−𝑦2)𝑗̇̂ + (𝑧1−𝑧2)𝑘̂
What is the correct representation of a position vector of point P with coordinates (-3, 2, 5)?
What is the correct representation of a position vector of point P with coordinates (-3, 2, 5)?
Which term symbolizes the change in position of an object?
Which term symbolizes the change in position of an object?
In the context of position vectors, what does the origin refer to?
In the context of position vectors, what does the origin refer to?
What happens to the position vector when an object's coordinates change?
What happens to the position vector when an object's coordinates change?
Which statement is true regarding the components of a position vector?
Which statement is true regarding the components of a position vector?
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Study Notes
Position Vector
- Defines the location of a point in space relative to a fixed reference point called the origin.
- Represented as a vector, denoted by 𝑟⃗.
- The position is a function of time, expressed as 𝑟⃗(𝑡).
- In a three-dimensional Cartesian coordinate system (x, y, z), the position vector for point P(x, y, z) relative to the origin O(0, 0, 0) is given by:
- 𝑟⃗ = 𝑥𝑖̇̂ + 𝑦𝑗̇̂ + 𝑧𝑘̂
- Example of a position vector: 𝑟⃗ = (−3𝑚)𝑖̇̂ + (2𝑚)𝑗̇̂ + (5𝑚)𝑘̂.
Displacement
- Defined as the change in position of an object from an initial position to a final position.
- Measures the relative change in position concerning a reference point.
- It is a vector quantity, indicating both magnitude and direction.
- Represented by the symbol 𝛥𝑟⃗.
- Displacement is calculated as:
- ∆𝑟⃗ = 𝑟⃗2 − 𝑟⃗1
- In unit-vector notation:
- ∆𝑟⃗ = (𝑥2𝑖̇̂ + 𝑦2𝑗̇̂ + 𝑧2𝑘̂) − (𝑥1𝑖̇̂ + 𝑦1𝑗̇̂ + 𝑧1𝑘̂)
- Simplified to: ∆𝑟⃗ = (𝑥2 − 𝑥1)𝑖̇̂ + (𝑦2 − 𝑦1)𝑗̇̂ + (𝑧2 − 𝑧1)𝑘̂
- Also expressed as ∆𝑟⃗ = ∆𝑥𝑖̇̂ + ∆𝑦𝑗̇̂ + ∆𝑧𝑘̂.
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