Spatial vectors, lines and angles · Vecteurs spatiaux, droites et angles
| English | Français |
|---|---|
| direction vector/daɪˈrekʃn ˈvektə/ | vecteur directeur |
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? · Quelle description définit correctement le vecteur directeur ?
A nonzero vector giving the direction of a line. · Un vecteur non nul donnant la direction d'une droite.
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 · Vecteurs spatiaux, droites et angles
A line has r=a+λb with position vector a and nonzero direction b · Une droite est donnée par r=a+λb avec vecteur position a et vecteur directeur b non nul.
Check all three components when deciding parallelism or perpendicularity. · Vérifier les trois composantes lors du choix entre parallélisme ou perpendicularité.
On r=(1,2,3)+λ(2,-1,2), find z at λ=2. · Sur r=(1,2,3)+λ(2,-1,2), trouvez z lorsque λ=2.
z=3+2λ; at λ=2 it is 7. · z=3+2λ ; lorsque λ=2, il vaut 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). · Trouvez la norme de (2,-1,2).
Magnitude=√(2²+(-1)²+2²)=3. · Norme=√(2²+(-1)²+2²)=3.
Any two nonparallel lines in three dimensions must intersect. · Deux droites non parallèles dans l'espace à trois dimensions doivent nécessairement se couper.
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. · Satisfaire deux équations de composantes ne garantit pas la troisième. Les droites non parallèles peuvent être gauches. Le vecteur directeur d'une droite n'est pas son vecteur position, et un paramètre scalaire n'a pas d'unités de coordonnées.
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). · Calculer (2,-1,2)·(1,0,-1).
Scalar product=2×1+(-1)×0+2×(-1)=0. · Produit scalaire=2×1+(-1)×0+2×(-1)=0.
Match each part of a complete solution to its purpose. · Associer chaque partie d'une solution complète à son but.
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · Une hypothèse justifie le modèle ; une vérification teste le résultat ; l'interprétation le relie à la question.
Use this in your course
- edexcel IAL pure mathematics; official unit P4. 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.