If $|\vec{a}| = 8$, $|\vec{b}| = 3$ and $|\vec{a} \times \vec{b}| = 12$, then the value of $|\vec{a} \cdot \vec{b}|$ is

Concept: $|a \times b| = |a||b|\sin\theta$ and $a \cdot b = |a||b|\cos\theta$, where theta is the angle between a and b.

Given $|a|=8$, $|b|=3$, $|a \times b|=12$: using $|a \times b|=|a||b|\sin\theta$, $12 = 24\sin\theta$, so $\sin\theta = \frac{1}{2}$, meaning theta = 30 degrees or 150 degrees. Using the Pythagorean identity, $\cos\theta = \pm \frac{\sqrt{3}}{2}$ in either case (positive for 30 degrees, negative for 150 degrees, but its magnitude is the same). So $|a \cdot b| = |a||b||\cos\theta| = 24 \times \frac{\sqrt{3}}{2} = 12\sqrt{3}$.

Why the distractors fail:

• (A) 6 sqrt(3) comes from using half of |a||b| (=12 instead of 24) in the final multiplication, an arithmetic slip.
• (B) 8 sqrt(3) mistakenly uses only |a| (=8) times sqrt(3) instead of the full product |a||b|cos(theta).
• (D) 3 sqrt(12) is a mathematically equal-looking but poorly simplified/incorrect combination (3 sqrt(12) = 6 sqrt(3), not 12 sqrt(3)), showing a common error of not simplifying surds correctly or mixing up which values multiply which.

Exam tip: Always find sin(theta) first from the cross product magnitude, get cos(theta) using sin-squared plus cos-squared equals 1, and only then substitute into the dot product formula — mixing up |a| and |b| values is the most common error here.