Vectors and transformation geometry
| English | 中文 | Pinyin |
|---|---|---|
| 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. A displacement from A to B is OB-OA. Parallel vectors are scalar multiples. A vector has magnitude and direction; a distance is a scalar.
- 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?
The vector sum representing the combined displacement or force.
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). Their scalar product is a·b=4×1+1×3=7. The magnitude of a is √17. For A=(1,2), B=(5,5), AB=(4,3) and |AB|=5.
Vectors and transformation geometry
Add corresponding vector components
Compare the model with the worked case and explain one change.
Find the x-component of (4,1)+(1,3).
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).
Magnitude=√(4²+3²)=5.
AB and BA always have the same components.
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
- For a translation, every point uses the same displacement. For advanced line problems, distinguish the position vector from a direction vector and solve parameter equations consistently.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the scalar product of (4,1) and (1,3).
Multiply corresponding components and add: 4×1+1×3=7.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- Current first-assessment-2021 Analysis and Approaches HL. This is authored concept support; the full guide is needed to certify every objective.
- 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.