Spatial vectors, lines and angles
| English | 中文 | Pinyin |
|---|---|---|
| direction vector/daɪˈrekʃn ˈvektə/ | 方向向量 | fāng xiàng xiàng liàng |
Can nonparallel paths still miss each other?
- Two paths in space can be neither parallel nor intersecting. A two-dimensional sketch may conceal their separation.
- This lesson studies direction vector 方向向量: A nonzero vector giving the direction of a line.
Choose the mathematical structure
- A line has r=a+λb with position vector a and nonzero direction b. To find an intersection, equate all components and solve for both line parameters. For an angle use a·b=|a||b|cosθ where scalar products are in the named syllabus.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines direction vector?
A nonzero vector giving the direction of a line.
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.
For a=(1,2,3),b=(2,-1,2), λ=2 gives r=(5,0,7). The magnitude of b is √(4+1+4)=3. With c=(1,0,-1), b·c=2-2=0, so b and c are perpendicular.
Spatial vectors, lines and angles
A line has r=a+λb with position vector a and nonzero direction b
Check all three components when deciding parallelism or perpendicularity.
On r=(1,2,3)+λ(2,-1,2), find z at λ=2.
z=3+2λ; at λ=2 it is 7.
Test a tempting shortcut
- Satisfying two component equations does not guarantee the third. Nonparallel lines may be skew. A line's direction vector is not its position vector, and a scalar parameter has no coordinate units.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Any two nonparallel lines in three dimensions must intersect. This claim is false. Explain which definition or assumption it violates.
Find the magnitude of (2,-1,2).
Magnitude=√(2²+(-1)²+2²)=3.
Any two nonparallel lines in three dimensions must intersect.
Satisfying two component equations does not guarantee the third. Nonparallel lines may be skew. A line's direction vector is not its position vector, and a scalar parameter has no coordinate units.
Interpret a new situation
- State whether a requested angle is acute, directed or an angle between lines. Use the syllabus-specific plane or distance method only when it is actually required for that unit.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find (2,-1,2)·(1,0,-1).
Scalar product=2×1+(-1)×0+2×(-1)=0.
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
- edexcel IAL further mathematics; official unit FP3. 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.
A nonzero vector giving the direction of a line. Choose the relationship, show the method, check its assumptions and interpret the result.