Vectors and transformation geometry
Введено| English | Значение |
|---|---|
| resultant/rɪˈzʌltənt/ | равнодействующий вектор |
Why is displacement shorter than the walk?
- Walking 4 m east and 3 m north gives a displacement of 5 m, even though the travelled distance is 7 m.
- This lesson studies resultant 合向量: The vector sum representing the combined displacement or force.
The method
- Add corresponding vector components and subtract position vectors to find a displacement.
- A translation moves every point by the same vector; a scalar multiple changes length and possibly direction.
What is the displacement vector from A=(1,2) to B=(5,5)?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
AB = OB - OA = (5,5)-(1,2) = (4,3). The order matters: BA would be (-4,-3), the opposite direction.
A worked case
With a=(4,1) and b=(1,3), a+b=(5,4). From A=(1,2) to B=(5,5), displacement AB=(4,3). Its magnitude is √(4²+3²)=5.
Vectors and transformation geometry
Add corresponding vector components and subtract position vectors to find a displacement
Compare the model with the worked case and explain one change.
Find the x-component of (4,1)+(1,3).
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Add the x-components: 4+1=5.
Where it goes wrong
- The order of subtraction matters: BA=-AB.
- Proving parallelism needs a scalar-multiple relation; a sketch alone is insufficient.
- Negative enlargement reverses position about its centre.
Not true: AB and BA have the same components. BA goes the opposite way to AB, so BA=−AB: every component changes sign.
Find the magnitude of (4,3).
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Magnitude=√(4²+3²)=5.
BA goes the opposite way to AB, so BA=−AB.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
True. BA goes the opposite way to AB, so BA=−AB: every component changes sign.
Applying it
- Use vector components, scalar multiples and magnitudes for displacements and forces.
- This unit does not require the scalar product.
Find the y-component of (4,1)+(1,3).
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Add the y-components: 1+3=4.
Match each vector property to its definition.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
A vector carries magnitude and direction; multiplying by a scalar keeps it parallel but changes its length or sense.
In your course
- AQA A-level Mathematics (7357), specification section J.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Resultant: The vector sum representing the combined displacement or force.
- Add corresponding vector components and subtract position vectors to find a displacement.
- Watch for: the order of subtraction matters: BA=-AB.