Vectors and transformation geometry · Vetores e geometria das transformações
| English | Português |
|---|---|
| resultant/rɪˈzʌltənt/ | resultante |
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.
Choose the mathematical structure
- 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.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines resultant? · Qual descrição define corretamente resultante?
The vector sum representing the combined displacement or force. · A soma vetorial que representa o deslocamento ou força combinados.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
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 · Vetores e geometria das transformações
Add corresponding vector components and subtract position vectors to find a displacement · Some as componentes vetoriais correspondentes e subtraia os vetores de posição para encontrar um deslocamento
Compare the model with the worked case and explain one change. · Compare o modelo com o caso resolvido e explique uma mudança.
Find the x-component of (4,1)+(1,3). · Encontre o componente x de (4,1)+(1,3).
Add the x-components: 4+1=5. · Some os componentes x: 4+1=5.
Test a tempting shortcut
- 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.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
AB and BA always have the same components. This claim is false. Explain which definition or assumption it violates.
Find the magnitude of (4,3). · Encontre a magnitude de (4,3).
Magnitude=√(4²+3²)=5.
AB and BA always have the same components. · AB e BA sempre têm os mesmos componentes.
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. · A ordem da subtração importa: BA=-AB. Comprovar paralelismo exige uma relação de múltiplo escalar; um esboço sozinho é insuficiente. Ampliação negativa inverte a posição em relação ao seu centro.
Interpret a new situation
- Use vector components, scalar multiples and magnitudes for displacements and forces. This unit does not require the scalar product.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the y-component of (4,1)+(1,3). · Encontre o componente y de (4,1)+(1,3).
Add the y-components: 1+3=4. · Some os componentes y: 1+3=4.
Match each part of a complete solution to its purpose. · Associe cada parte de uma solução completa ao seu propósito.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Uma suposição justifica o modelo; uma verificação testa o resultado; a interpretação conecta-o à pergunta.
Use this in your course
- edexcel IAL further mathematics; official unit M1. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
The vector sum representing the combined displacement or force. Choose the relationship, show the method, check its assumptions and interpret the result.