Vector product
A product that gives a vector
You know the dot product: multiply two vectors and get a number. Now meet a product that gives a new vector.
Take two vectors a and b, tail to tail, with angle θ between them. They lie flat in a plane, like two pencils on a table.
Their vector product is written a × b and read “a cross b”. Its size is ab sin θ.
Look at the parallelogram formed by a and b. Its area is also ab sin θ. So the size of a × b equals this area.
Its direction is perpendicular to the plane of a and b: straight up out of the table, or straight down into it.
To choose, use the right-hand rule. Point your right-hand fingers along a, then curl them towards b through the smaller angle. Your thumb points along a × b.
Or picture a screw: turn it from a towards b, and it moves in the direction of a × b.
When θ is 0° or 180°, sin θ is zero, so parallel vectors have zero cross product. At θ = 90° the product is as big as it can be: ab.
Rules of the vector product
Swap the order. Curl your fingers from b to a, and your thumb flips over. So b × a = −a × b. The vector product is not commutative.
Any vector crossed with itself gives zero: a × a = 0, because the angle is zero.
For the unit vectors along x, y and z: î × î = ĵ × ĵ = k̂ × k̂ = 0.
Going round the cycle x, y, z: î × ĵ = k̂, ĵ × k̂ = î and k̂ × î = ĵ. Go the other way and you get a minus sign: ĵ × î = −k̂.
The vector product is distributive: a × (b + c) = a × b + a × c. So we can multiply out components term by term.
That gives the determinant form. Write î, ĵ, k̂ in the first row, the components of a in the second row and of b in the third, then expand.
a × b = (a_y b_z − a_z b_y) î + (a_z b_x − a_x b_z) ĵ + (a_x b_y − a_y b_x) k̂.
Torque and angular momentum are both defined with this product, so it is worth practising now.
Explore it in 3D
- Drag to orbit the view. Use the sliders to turn a and b in the x–y plane.
- Press “Swap order”: the arrow flips, but its length stays the same.
- Set the two directions equal, or 180° apart. Where did the product go?
Quick check
Two non-zero vectors a and b have a × b = 0. What can you say about them?
- They are perpendicular
- They are parallel or opposite
- They have equal size
- One of them must be a unit vector
Answer
B. |a × b| = ab sin θ. With a and b not zero, the product can only vanish if sin θ = 0, so θ = 0° or 180°. Perpendicular vectors give the biggest cross product.
Work it out
Find (2î + 3ĵ) × (î − ĵ).
- 5k̂
- −5k̂
- −k̂
- 2î − 3ĵ
Answer
B. Multiply out: 2î × î = 0, 2î × (−ĵ) = −2k̂, 3ĵ × î = −3k̂, 3ĵ × (−ĵ) = 0. Total −5k̂. Check with the formula: a_x b_y − a_y b_x = 2(−1) − 3(1) = −5.
Summary
- Definition:
a × b = ab sin θ n̂ - Order matters:
b × a = −a × b - Unit vectors:
î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ - Components:
a × b = (a_y b_z − a_z b_y) î + (a_z b_x − a_x b_z) ĵ + (a_x b_y − a_y b_x) k̂
- a × b is perpendicular to both a and b. Its direction comes from the right-hand rule.
- Its size, ab sin θ, is the area of the parallelogram formed by a and b.
- Parallel or opposite vectors give zero; perpendicular vectors give the largest value, ab.
- Swapping the order flips the sign, and a × a = 0.