Question

If $\frac{\pi}{2} < \theta < \pi$ and $|\bar{a}| = 5$, $|\bar{b}| = 13$, $|\bar{a} \times \bar{b}| = 25$, then the value of $\bar{a} \cdot \bar{b}$ is

• A. -12
• B. 60
• C. -60
• D. -13

Hint:

You can solve this even faster using Lagrange's identity:
$$( \bar{a} \cdot \bar{b} )^2 + |\bar{a} \times \bar{b}|^2 = |\bar{a}|^2 |\bar{b}|^2$$ $$( \bar{a} \cdot \bar{b} )^2 + 25^2 = (5 \times 13)^2 \implies ( \bar{a} \cdot \bar{b} )^2 + 625 = 4225$$ $$( \bar{a} \cdot \bar{b} )^2 = 3600 \implies \bar{a} \cdot \bar{b} = \pm 60$$ Since $\theta$ is in the second quadrant, the dot product must be negative, giving $-60$ instantly.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given the magnitudes of two vectors $\bar{a}$ and $\bar{b}$, along with the magnitude of their cross product. The angle $\theta$ between the vectors lies strictly within the second quadrant ($90^\circ < \theta < 180^\circ$). We need to determine the value of their dot product $\bar{a} \cdot \bar{b}$.

Step 2: Key Formula or Approach:
We use the fundamental geometric definitions of vector multiplication:
1. Cross product magnitude: $|\bar{a} \times \bar{b}| = |\bar{a}| |\bar{b}| \sin\theta$
2. Dot product value: $\bar{a} \cdot \bar{b} = |\bar{a}| |\bar{b}| \cos\theta$
We will use the cross product data to solve for $\sin\theta$, find $\cos\theta$ keeping the quadrant sign constraints in mind, and then compute the dot product.

Step 3: Detailed Explanation:
Substitute the given values into the cross product magnitude formula:
$$25 = 5 \times 13 \times \sin\theta$$ $$25 = 65 \sin\theta \implies \sin\theta = \frac{25}{65} = \frac{5}{13}$$ We know the trigonometric identity: $\cos^2\theta = 1 - \sin^2\theta$.
$$\cos^2\theta = 1 - \left(\frac{5}{13}\right)^2 = 1 - \frac{25}{169} = \frac{144}{169}$$ Taking the square root gives: $\cos\theta = \pm \frac{12}{13}$.
Since the problem explicitly specifies that $\frac{\pi}{2} < \theta < \pi$ (Quadrant II), the cosine function must take a negative value:
$$\cos\theta = -\frac{12}{13}$$ Now, calculate the final scalar dot product value:
$$\bar{a} \cdot \bar{b} = |\bar{a}| |\bar{b}| \cos\theta = 5 \times 13 \times \left(-\frac{12}{13}\right)$$ The factor of 13 cancels out perfectly:
$$\bar{a} \cdot \bar{b} = 5 \times (-12) = -60$$

Step 4: Final Answer:
The dot product value is -60, which corresponds to option (C).