If θ is the angle between vectors \(\vec{a}\) and \( \vec{b}\) such that \(\vec{a} \cdot \vec{b} \geq 0\), then which one of the following is correct?
Finding the Angle Range from Vector Dot Product Condition
The question asks about the relationship between the angle \(\theta\) between two vectors \(\vec{a}\) and \(\vec{b}\) and the condition that their dot product is non-negative, i.e., \(\vec{a} \cdot \vec{b} \geq 0\). We need to determine the possible range for the angle \(\theta\).
The dot product of two vectors \(\vec{a}\) and \(\vec{b}\) is defined as:
\(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta\)
where \(|\vec{a}|\) is the magnitude of vector \(\vec{a}\), \(|\vec{b}|\) is the magnitude of vector \(\vec{b}\), and \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\). The angle \(\theta\) is usually considered to be in the range \(0 \leq \theta \leq \pi\) radians (or \(0^\circ \leq \theta \leq 180^\circ\)).
We are given the condition:
\(\vec{a} \cdot \vec{b} \geq 0\)
Substituting the definition of the dot product, we get:
\(|\vec{a}| |\vec{b}| \cos \theta \geq 0\)
Let's analyze this inequality.
- The magnitudes \(|\vec{a}|\) and \(|\vec{b}|\) are always non-negative (\(\geq 0\)).
- If either \(\vec{a}\) or \(\vec{b}\) is the zero vector, then \(|\vec{a}| = 0\) or \(|\vec{b}| = 0\), which makes \(|\vec{a}| |\vec{b}| = 0\). In this case, \(\vec{a} \cdot \vec{b} = 0\), and the condition \(\vec{a} \cdot \vec{b} \geq 0\) is satisfied. The angle between a non-zero vector and the zero vector is typically undefined. However, the options provide a specific range for \(\theta\), suggesting we are dealing with cases where the angle is well-defined. This usually implies non-zero vectors.
- If both \(\vec{a}\) and \(\vec{b}\) are non-zero vectors, then \(|\vec{a}| > 0\) and \(|\vec{b}| > 0\). This means \(|\vec{a}| |\vec{b}| > 0\).
Assuming \(\vec{a}\) and \(\vec{b}\) are non-zero vectors, we can divide the inequality \(|\vec{a}| |\vec{b}| \cos \theta \geq 0\) by the positive value \(|\vec{a}| |\vec{b}|\). This gives us:
\(\cos \theta \geq 0\)
Now we need to find the range of \(\theta\) (within the standard range \(0 \leq \theta \leq \pi\)) for which \(\cos \theta\) is greater than or equal to zero.
Consider the graph of the cosine function or the unit circle. In the range \(0 \leq \theta \leq \pi\):
- \(\cos \theta = 1\) at \(\theta = 0\).
- \(\cos \theta\) decreases as \(\theta\) increases from 0 to \(\pi\).
- \(\cos \theta = 0\) at \(\theta = \frac{\pi}{2}\).
- \(\cos \theta\) is positive for \(0 \leq \theta < \frac{\pi}{2}\).
- \(\cos \theta\) is negative for \(\frac{\pi}{2} < \theta \leq \pi\).
So, the condition \(\cos \theta \geq 0\) is satisfied when \(\theta\) is in the range \(0 \leq \theta \leq \frac{\pi}{2}\).
This means if the dot product of two non-zero vectors is non-negative, the angle between them must be between 0 and \(\frac{\pi}{2}\) (inclusive). This corresponds to an acute angle or a right angle.
Comparing this derived range with the given options:
| Option | Angle Range |
|---|---|
| 1 | \(0 \leq \theta \leq \pi\) |
| 2 | \(\frac{\pi}{2} \leq \theta \leq \pi\) |
| 3 | \(0 \leq \theta \leq \frac{\pi}{2}\) |
| 4 | \(0 < \theta < \frac{\pi}{2}\) |
The range \(0 \leq \theta \leq \frac{\pi}{2}\) matches Option 3.
If one or both vectors are the zero vector, \(\vec{a} \cdot \vec{b} = 0\), which satisfies \(\vec{a} \cdot \vec{b} \geq 0\). While the angle is typically undefined, if we were to consider it, any angle \(\theta\) wouldn't contradict the dot product being 0 based on the formula \(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta\). However, the options suggest a specific range for \(\theta\) which arises directly from the \(\cos \theta \geq 0\) condition derived for non-zero vectors. The most common interpretation of "angle between vectors" when given options like these pertains to non-zero vectors.
Thus, for the angle \(\theta\) between vectors \(\vec{a}\) and \(\vec{b}\), the condition \(\vec{a} \cdot \vec{b} \geq 0\) implies \(0 \leq \theta \leq \frac{\pi}{2}\).
Revision Table: Key Concepts for Vector Angles
| Concept | Description | Condition | Angle Range (\(0 \leq \theta \leq \pi\)) |
|---|---|---|---|
| Dot Product | \(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta\) | N/A | \(0 \leq \theta \leq \pi\) |
| Dot Product is Positive | Vectors point generally in the same direction (Acute angle) | \(\vec{a} \cdot \vec{b} > 0\) | \(0 \leq \theta < \frac{\pi}{2}\) |
| Dot Product is Zero | Vectors are orthogonal (Perpendicular) | \(\vec{a} \cdot \vec{b} = 0\) | \(\theta = \frac{\pi}{2}\) |
| Dot Product is Negative | Vectors point generally in opposite directions (Obtuse angle) | \(\vec{a} \cdot \vec{b} < 0\) | \(\frac{\pi}{2} < \theta \leq \pi\) |
| Dot Product is Non-Negative | Vectors point generally in the same direction or are orthogonal | \(\vec{a} \cdot \vec{b} \geq 0\) | \(0 \leq \theta \leq \frac{\pi}{2}\) |
| Dot Product is Non-Positive | Vectors point generally in opposite directions or are orthogonal | \(\vec{a} \cdot \vec{b} \leq 0\) | \(\frac{\pi}{2} \leq \theta \leq \pi\) |
Additional Information on Vector Angles and Dot Product
The dot product (also known as the scalar product) is a fundamental operation in vector algebra that takes two vectors and returns a scalar quantity. Its value is directly related to the magnitudes of the vectors and the cosine of the angle between them.
- The geometric interpretation of the dot product is particularly useful for understanding the relationship with the angle. \(\vec{a} \cdot \vec{b}\) is the product of the magnitude of \(\vec{a}\) and the scalar projection of \(\vec{b}\) onto \(\vec{a}\) (or vice versa).
- When \(\vec{a} \cdot \vec{b} > 0\), the projection of one vector onto the other points in the same direction as the first vector, indicating an acute angle (\(0 \leq \theta < \frac{\pi}{2}\)).
- When \(\vec{a} \cdot \vec{b} = 0\), the projection is zero, meaning the vectors are perpendicular or orthogonal (\(\theta = \frac{\pi}{2}\)).
- When \(\vec{a} \cdot \vec{b} < 0\), the projection points in the opposite direction, indicating an obtuse angle (\(\frac{\pi}{2} < \theta \leq \pi\)).
The condition \(\vec{a} \cdot \vec{b} \geq 0\) combines the cases where the dot product is positive or zero. Geometrically, this means the angle is either acute or a right angle.
Understanding the relationship between the sign of the dot product and the range of the angle is crucial for solving problems involving vector geometry and mechanics.
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