Mathematics

Mathematics

Problem

A vector P⃗\vec P points due north (along +j^+\hat j) and another vector Q⃗\vec Q points due east (along +i^+\hat i). The direction of P⃗×Q⃗\vec P \times \vec Q is a) along +i^+\hat i b) along −k^-\hat k (downward) c) along +k^+\hat k (upward) d) along +j^+\hat j
Intuitive explanation

The cross product P⃗×Q⃗\vec P \times \vec Q follows the right-hand rule: curl your fingers from the first vector toward the second, and your thumb points in the direction of the result. Since P⃗\vec P points north and Q⃗\vec Q points east, rotating from north toward east (a clockwise turn when viewed from above) gives a direction pointing downward—perpendicular to both original vectors. This is the key insight: the cross product is always perpendicular to both inputs, and the right-hand rule tells you which perpendicular direction.

Intuitive explanation

The cross product P⃗×Q⃗\vec P \times \vec Q follows the right-hand rule: curl your fingers from the first vector toward the second, and your thumb points in the direction of the result. Since P⃗\vec P points north and Q⃗\vec Q points east, rotating from north toward east (a clockwise turn when viewed from above) gives a direction pointing downward—perpendicular to both original vectors. This is the key insight: the cross product is always perpendicular to both inputs, and the right-hand rule tells you which perpendicular direction.

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