Practice Flashcards
Flip to reveal answersWhat is |v×w| in terms of the angle θ?
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All 8 Flashcards — Length of v×w and areas
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Question
What is |v×w| in terms of the angle θ?
Answer
|v×w| = |v||w| sin θ (the dot product used cos θ; the cross uses sin θ).
Question
Area of the parallelogram with sides v and w?
Answer
|v×w| (the length of the cross product).
Question
Area of the triangle with sides v and w?
Answer
½|v×w| (half the parallelogram).
Question
How do you find the area of triangle ABC with the cross product?
Answer
Form AB and AC, compute AB×AC, take its length, then halve: ½|AB×AC|.
Question
State the identity linking the cross and dot products.
Answer
|v×w|² = |v|²|w|² − (v·w)².
Question
Why is |v×w| = 0 when v and w are parallel?
Answer
θ = 0 ⇒ sin θ = 0, so the length is 0 (zero parallelogram area).
Question
|v×w| when |v| = 5, |w| = 4, θ = 30°?
Answer
5·4·sin 30° = 20·½ = 10.
Question
Find the triangle area if |v| = 3, |w| = 5, v·w = 9.
Answer
|v×w|² = 9·25 − 81 = 144, |v×w| = 12, area = ½·12 = 6.
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Full study notes for Length of v×w and areas
Topic 3.16 hub
Vector product (HL only)
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