g10-08-03 · Syllabus supplement
A negative scalar reverses direction
What does multiplication by−2 change?
1 MIN · PREDICT ON PAPER
Compare a,2a,−2a and0a for a=(2,1).
Discuss or write first. No typed answer is required to continue.
Magnitude uses |λ|; direction comparison excludes zero.
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
λa has magnitude |λ||a|; λ<0 reverses direction. A zero product has no defined direction.
Conditions: Direction comparisons require both vectors to be nonzero. Scalar multiples are collinear algebraically, including a zero product.
Mathematical justification: The components become(λa₁,λa₂), whose norm is√(λ²(a₁²+a₂²))=|λ||a|.
1 MIN · EXIT CHECK
Find−3(1,−2) and its magnitude.
Check after answering
(−3,6), magnitude3√5.
Teacher guidance and extension
Before: Coordinates, displacement and Pythagoras.
Ask: What changes sign: the components or the length?
Watch for: Saying a negative multiple has a negative length.
Challenge: Explain why0a cannot be called opposite in direction to a.
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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