g10-08-05 · Syllabus supplement
What does the sign of a dot product tell us?
Does a dot product of0 always mean a right angle?
1 MIN · PREDICT ON PAPER
Compare two nonzero vectors at60°,90°,120°, then let one length become0.
Discuss or write first. No typed answer is required to continue.
Positive dot product: acute angle.
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
For nonzero vectors, a·b=|a||b|cosθ is positive,zero or negative for acute,right or obtuse angles. With a zero vector the angle is undefined.
Conditions: Both vectors must be nonzero before concluding perpendicularity from a zero dot product.
Mathematical justification: The lengths are strictly positive for nonzero vectors, so the dot-product sign equals the sign of cosθ. At θ=90°,cosθ=0.
1 MIN · EXIT CHECK
What extra condition is needed in “a·b=0 implies a⊥b”?
Check after answering
a andb must both be nonzero.
Teacher guidance and extension
Before: Coordinates, displacement and Pythagoras.
Ask: Which assumption lets us divide the dot product by both lengths?
Watch for: Declaring a zero vector perpendicular to every vector because its dot products vanish.
Challenge: Produce a nonzero pair with negative dot product.
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.
Optional local notebook
Paper responses are enough. Drafts remain in this tab; choose local saving to include them in the existing complete backup.
Complete backup and restore