Vectors and transformation geometry · Foundation
導入済み| English | 意味 |
|---|---|
| resultant/rɪˈzʌltənt/ | 合成ベクトル |
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.
With a=(4,1) and b=(1,3), what is a+b?
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
Add corresponding components: (4+1, 1+3)=(5,4). Each component is added separately.
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
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).
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Add the x-components: 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).
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Magnitude=√(4²+3²)=5.
AB and BA always have the same components.
難易度 1/5 · 100 コイン + 今日初回完了時に小ボーナス付与
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.
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).
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Add the y-components: 1+3=4.
Match each operation to its result on a vector.
難易度 2/5 · 140 コイン + 今日初回完了時に小ボーナス付与
Vector arithmetic acts component by component; scalars change length and possibly direction.
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.