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. 1
• 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: Recall the formula for the dot product of two vectors:

The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is defined as:

\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos \theta \]

Where:

• \( \mathbf{a} \) and \( \mathbf{b} \) represent the two vectors.
• \( |\mathbf{a}| \) and \( |\mathbf{b}| \) denote the magnitudes of the respective vectors.
• \( \theta \) is the angle situated between the two vectors.

Step 2: Substitute the known values into the formula:

Given that the angle between the vectors is \( \theta = 60^\circ \), we substitute this value into the formula:

\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos 60^\circ \]

As \( \cos 60^\circ = \frac{1}{2} \), the dot product simplifies to:

\[ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \times \frac{1}{2} \]

Step 3: Determine the condition for a positive dot product:

The dot product \( \mathbf{a} \cdot \mathbf{b} \) yields a positive value when the cosine of the angle is positive. For \( \theta = 60^\circ \), where \( \cos \theta = \frac{1}{2} \), the dot product is positive because the product of the magnitudes, \( |\mathbf{a}| |\mathbf{b}| \), is inherently positive (assuming non-zero vectors). Consequently, the dot product \( \mathbf{a} \cdot \mathbf{b} \) is positive when \( \theta = 60^\circ \).

Conclusion:

The probability that the dot product \( \mathbf{a} \cdot \mathbf{b} \) is positive is \( \mathbf{1} \), as the dot product is invariably positive for an angle of \( 60^\circ \).