Question

If \( \mathbf{a} \) and \( \mathbf{b} \) are two non-zero vectors such that the angle between them is \( 60^\circ \), what is the probability that the dot product \( \mathbf{a} \cdot \mathbf{b} \) is positive?

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{3} \)
• C. \( \frac{2}{3} \)
• D. \( \frac{1}{4} \)

Hint:

Remember that the dot product of two vectors is positive when the angle between them is less than \( 90^\circ \). The probability can be calculated by comparing the favorable angle range with the total possible angle range.

Answer 1

The Correct Option is C

Step 1: Formula for the dot product. The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by: \[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \] where \( \theta \) is the angle between the vectors. Step 2: Conditions for positive dot product. For the dot product \( \mathbf{a} \cdot \mathbf{b} \) to be positive, we need \( \cos \theta>0 \). This will occur when the angle \( \theta \) is between \( 0^\circ \) and \( 90^\circ \), as \( \cos \theta \) is positive in this range. Step 3: Considering the full range of angles. The angle \( \theta \) between two vectors can range from \( 0^\circ \) to \( 180^\circ \). Thus, the total possible range for the angle \( \theta \) is \( 180^\circ \). The dot product is positive when \( 0^\circ < \theta < 90^\circ \), which is a range of \( 90^\circ \). Step 4: Probability calculation. The probability that the dot product is positive is the ratio of the favorable range (where \( \cos \theta > 0 \)) to the total possible range of angles: \[ \text{Probability} = \frac{90^\circ}{180^\circ} = \frac{1}{2} \] Answer: Therefore, the probability that the dot product \( \mathbf{a} \cdot \mathbf{b} \) is positive is \( \frac{1}{2} \).