g10-08-04 · Syllabus supplement
Two directions need to be independent
Can any pair of vectors locate every point uniquely?
1 MIN · PREDICT ON PAPER
Keep e₁=(1,0). Compare e₂=(0,1) ande₂=(1,0) for target(3,2).
Discuss or write first. No typed answer is required to continue.
The independent directions give one coefficient pair.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
Two noncollinear vectors form a basis and give unique coefficients. Collinear vectors cover only a line, and may give no solution or nonunique solutions.
Conditions: The determinant e₁x e₂y−e₁y e₂x must be nonzero for a plane basis.
Mathematical justification: The two component equations have a unique solution precisely when their2×2 determinant is nonzero. A zero determinant makes the directions dependent.
1 MIN · EXIT CHECK
Do(1,0) and(2,0) form a plane basis?
Check after answering
No; they are collinear and cannot reach(0,1).
Teacher guidance and extension
Before: Coordinates, displacement and Pythagoras.
Ask: What region can be reached when the two arrows point along the same line?
Watch for: Assuming two arrows are a basis simply because there are two of them.
Challenge: Give a target on the common line with more than one coefficient pair.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
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