Practice Flashcards
What kind of object is the cross product v×w?
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All Flashcards in Topic 3.16
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3.16.18 cards
What kind of object is the cross product v×w?
A vector (in 3D), unlike the dot product v·w which is a number.
Geometrically, where does v×w point?
Perpendicular to BOTH v and w — straight out of the plane they span.
Write the determinant formula for v×w.
v×w = |i j k; v₁ v₂ v₃; w₁ w₂ w₃| = (v₂w₃−v₃w₂, v₃w₁−v₁w₃, v₁w₂−v₂w₁).
Which component of the cross product carries a built-in minus sign?
The middle (j) component: v₃w₁ − v₁w₃ (i.e. −(v₁w₃ − v₃w₁)).
How does w×v relate to v×w?
w×v = −(v×w): swapping the order reverses every component (anti-commutative).
Find i×j (axis unit vectors).
i×j = k (points along the z-axis).
Quick check that v×w is correct?
Dot it with v (or w): v·(v×w) should be 0, since v×w ⊥ v.
Find v×w for v = (2, 3, 1), w = (1, −1, 4).
(3·4−1·(−1), 1·1−2·4, 2·(−1)−3·1) = (13, −7, −5).
3.16.28 cards
What is |v×w| in terms of the angle θ?
|v×w| = |v||w| sin θ (the dot product used cos θ; the cross uses sin θ).
Area of the parallelogram with sides v and w?
|v×w| (the length of the cross product).
Area of the triangle with sides v and w?
½|v×w| (half the parallelogram).
How do you find the area of triangle ABC with the cross product?
Form AB and AC, compute AB×AC, take its length, then halve: ½|AB×AC|.
State the identity linking the cross and dot products.
|v×w|² = |v|²|w|² − (v·w)².
Why is |v×w| = 0 when v and w are parallel?
θ = 0 ⇒ sin θ = 0, so the length is 0 (zero parallelogram area).
|v×w| when |v| = 5, |w| = 4, θ = 30°?
5·4·sin 30° = 20·½ = 10.
Find the triangle area if |v| = 3, |w| = 5, v·w = 9.
|v×w|² = 9·25 − 81 = 144, |v×w| = 12, area = ½·12 = 6.
Topic 3.16 study notes
Full notes & explanations for Vector product (HL only)
Math AA exam skills
Paper structures, command terms & tips
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