Vectors and transformation geometry · Foundation · 向量与变换几何 · 基础
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| resultant/rɪˈzʌltənt/ | 合向量 | hé xiàng liàng |
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? · 哪个描述正确定义了合力？
The vector sum representing the combined displacement or force. · 表示合位移或合力的向量和。
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) and 2a-b=(7,-1). From A=(1,2) to B=(5,5), AB=(4,3) and its length is 5. Write each 2D vector as a column with the horizontal component above the vertical component: the translation AB has top entry 4 and bottom entry 3. A negative horizontal entry moves left; a negative vertical entry moves down. Drawing b from the head of a gives the head-to-tail diagram for a+b. Subtracting b adds its opposite -b.
Vectors and transformation geometry · 向量与变换几何
Add corresponding vector components and subtract position vectors to find a displacement · 相加对应的向量分量并相减位置矢量以找到位移。
Compare the model with the worked case and explain one change. · 对比模型与已解案例，并说明其中一处变化。
Find the x-component of (4,1)+(1,3). · 求向量 (4,1)+(1,3) 的 x 分量。
Add the x-components: 4+1=5. · 将x分量相加：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). · 求(4,3)的模长。
Magnitude=√(4²+3²)=5. · 模长=√(4²+3²)=5。
AB and BA always have the same components. · AB和BA始终具有相同的分量。
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. · 减法的顺序很重要：BA=-AB。证明平行需要标量倍数关系；仅凭草图是不够的。负位似变换会使其关于中心反向。
Interpret a new situation
- Draw arrows with direction and identify the starting and ending points. Scalar products, spatial line equations and advanced angle calculations are excluded from this Foundation/Core lesson.
- 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). · 求(4,1)+(1,3)的y分量。
Add the y-components: 1+3=4. · 将y分量相加：1+3=4。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- 8300 · Foundation · 3.4. Match the target tier and specification before assigning extensions.
- 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.