Mathematics

Mathematics

Problem

8. Find the magnitude of two vectors a⃗\vec{a} and b⃗\vec{b}, having the same magnitude and such that the angle between them is 60∘60^{\circ} and their scalar product is 12\frac{1}{2}.
Intuitive explanation

The key is recognizing that the scalar (dot) product formula a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta directly connects what you know—the angle and the dot product value—to what you're finding: the magnitude. Since both vectors have equal magnitude, you can set ∣a⃗∣=∣b⃗∣=m|\vec{a}| = |\vec{b}| = m and substitute into the formula to solve for mm.

Intuitive explanation

The key is recognizing that the scalar (dot) product formula a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta directly connects what you know—the angle and the dot product value—to what you're finding: the magnitude. Since both vectors have equal magnitude, you can set ∣a⃗∣=∣b⃗∣=m|\vec{a}| = |\vec{b}| = m and substitute into the formula to solve for mm.

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