Question

If $\theta$ is an obtuse angle between vectors $\vec{a}$ and $\vec{b}$ such that $|\vec{a}| = 5, |\vec{b}| = 3$ and $|\vec{a} \times \vec{b}| = 5\sqrt{5}$ then $\vec{a} \cdot \vec{b} = \dots$

• A. 10
• B. -10
• C. 5
• D. -5

Hint:

Alternatively, use Lagrange's Identity: $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2$.
$(5\sqrt{5})^2 + (\vec{a} \cdot \vec{b})^2 = (25)(9) \implies 125 + x^2 = 225 \implies x^2 = 100 \implies x = \pm 10$. Choose the negative root because the angle is obtuse!

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We use the relation between the cross product and dot product: $|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2 |\vec{b}|^2$ (Lagrange's Identity).

Step 2: Formula Application: $(5\sqrt{5})^2 + (\vec{a} \cdot \vec{b})^2 = (5)^2 (3)^2$. $125 + (\vec{a} \cdot \vec{b})^2 = 25 \times 9 = 225$.

Step 3: Explanation:
$(\vec{a} \cdot \vec{b})^2 = 225 - 125 = 100 \implies \vec{a} \cdot \vec{b} = \pm 10$. Since $\theta$ is an obtuse angle ($90^\circ < \theta < 180^\circ$), $\cos \theta$ is negative. Therefore, $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos \theta$ must be negative. $\vec{a} \cdot \vec{b} = -10$.

Step 4: Final Answer:
The dot product is -10.