Product of two vectors

Physics · Vectors and equilibrium. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Two vectors can be multiplied in two ways. The scalar product (dot product) gives a number: it multiplies one vector by the part of the other that lies along it. It is largest when the vectors are parallel and zero when they are perpendicular. Work and power are scalar products.

The vector product (cross product) gives a new vector that is perpendicular to both of the original vectors. Its direction comes from the right-hand rule: curl the fingers of the right hand from the first vector to the second through the smaller angle, and the thumb gives the direction. It is zero for parallel vectors and largest for perpendicular ones. Torque is a vector product.

Rules to remember

A · B = AB cos θ = A_xB_x + A_yB_y + A_zB_z.
A · B = B · A; i · i = j · j = k · k = 1; i · j = j · k = k · i = 0.
Angle between vectors: cos θ = (A · B) ÷ (AB).
Magnitude of A × B = AB sin θ = area of the parallelogram with A and B as sides.
A × B = −(B × A); i × j = k, j × k = i, k × i = j; i × i = j × j = k × k = 0.
Work W = F · d; power P = F · v; torque τ = r × F.

Common mistake

Students treat the cross product like ordinary multiplication and write j × i = k. Changing the order changes the sign, so j × i = −k.

Worked example

A = 2i + 3j and B = i + 4j. Find A × B.

Multiply term by term: A × B = 2(i × i) + 8(i × j) + 3(j × i) + 12(j × j).
Use i × i = 0, j × j = 0, i × j = k and j × i = −k.
A × B = 0 + 8k − 3k + 0 = 5k.

Answer: A × B = 5k, a vector of magnitude 5 units along the positive z-axis

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