The cross product by components
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Flip to reveal answersWhat kind of object is the cross product v×w?
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All 8 Flashcards — The cross product by components
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Question
What kind of object is the cross product v×w?
Answer
A vector (in 3D), unlike the dot product v·w which is a number.
Question
Geometrically, where does v×w point?
Answer
Perpendicular to BOTH v and w — straight out of the plane they span.
Question
Write the determinant formula for v×w.
Answer
v×w = |i j k; v₁ v₂ v₃; w₁ w₂ w₃| = (v₂w₃−v₃w₂, v₃w₁−v₁w₃, v₁w₂−v₂w₁).
Question
Which component of the cross product carries a built-in minus sign?
Answer
The middle (j) component: v₃w₁ − v₁w₃ (i.e. −(v₁w₃ − v₃w₁)).
Question
How does w×v relate to v×w?
Answer
w×v = −(v×w): swapping the order reverses every component (anti-commutative).
Question
Find i×j (axis unit vectors).
Answer
i×j = k (points along the z-axis).
Question
Quick check that v×w is correct?
Answer
Dot it with v (or w): v·(v×w) should be 0, since v×w ⊥ v.
Question
Find v×w for v = (2, 3, 1), w = (1, −1, 4).
Answer
(3·4−1·(−1), 1·1−2·4, 2·(−1)−3·1) = (13, −7, −5).
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Full study notes for The cross product by components
Topic 3.16 hub
Vector product (HL only)
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