Two vectors in, a perpendicular vector out: Hold two pencils meeting at a point — they span a flat sheet. The cross product v×w is a brand-new vector that sticks straight out of that sheet, at right angles to both pencils.
Key contrast: the dot product v·w gives a number; the cross product v×w gives a vector. So v×w has both a direction (out of the plane) and a length (you'll meet its length in 3.16.2).
Only in 3D: The cross product is defined for three-dimensional vectors only. The result lives in 3D too, and (by its right-angle property) it can never lie flat inside the plane of v and w.
IB-style question — cross product of the axis vectors
Let i, j, k be the unit vectors along the x-, y-, z-axes.
Find i×j and describe its direction.
Step by step
- Write i = (1, 0, 0) and j = (0, 1, 0) and apply the rule (built in §2).
- Simplify each component.
- i and j lie in the horizontal (x-y) plane, so their cross product points straight up the z-axis.
Final answer
i×j = k — perpendicular to both, pointing along the positive z-axis.
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The determinant pattern: For v = (v₁, v₂, v₃) and w = (w₁, w₂, w₃), the components of v×w come from a 3×3 determinant. Each component leaves out its own row and cross-multiplies the other two — with a minus sign on the middle (j) component.
A reliable hand-method: For each component, cover its own row and cross-multiply the leftover 2×2 block (top-left × bottom-right − top-right × bottom-left):
1st (i): rows 2,3 → v₂w₃ − v₃w₂.
2nd (j): v₃w₁ − v₁w₃ (this is −(v₁w₃ − v₃w₁), so the sign is already handled).
3rd (k): v₁w₂ − v₂w₁.
IB-style question — cross product by components
Let v = (2, 3, 1) and w = (1, −1, 4).
Find v×w.
Step by step
- First (i) component: v₂w₃ − v₃w₂.
- Second (j) component: v₃w₁ − v₁w₃.
- Third (k) component: v₁w₂ − v₂w₁.
- Assemble the vector.
Final answer
v×w = (13, −7, −5).
IB-style question — order matters
Using v = (2, 3, 1) and w = (1, −1, 4) from above, find w×v.
Step by step
- Swapping the order reverses every component: w×v = −(v×w).
- Negate each component of v×w = (13, −7, −5).
Final answer
w×v = (−13, 7, 5) — the same line but opposite direction (anti-commutative).