Problem 43
Question
If \(\mathbf{a} \cdot \mathbf{b}=\sqrt{3}\) and \(\mathbf{a} \times \mathbf{b}=\langle 1,2,2\rangle,\) find the angle between a and \(\mathbf{b} .\)
Step-by-Step Solution
Verified Answer
The angle between \(\mathbf{a}\) and \(\mathbf{b}\) is 60 degrees or \(\frac{\pi}{3}\) radians.
1Step 1: Recall Key Dot and Cross Product Formulas
Remember that the dot product formula for vectors \(\mathbf{a}\) and \(\mathbf{b}\) is \(\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta\), where \(\theta\) is the angle between the vectors. The magnitude of the cross product is given by \(\|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin\theta\).
2Step 2: Calculate Magnitude of Cross Product
Using the given cross product \(\mathbf{a} \times \mathbf{b} = \langle 1, 2, 2 \rangle\), compute its magnitude: \[\|\mathbf{a} \times \mathbf{b}\| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{9} = 3.\]
3Step 3: Form Equations for Dot and Cross Products
Using the established formulas, set up two equations: \(\mathbf{a} \cdot \mathbf{b} = \sqrt{3} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta\) and \(3 = \|\mathbf{a}\| \|\mathbf{b}\| \sin\theta\).
4Step 4: Divide the Magnitude Equations
Divide the equation for the magnitude of the cross product by the equation for the dot product: \[\frac{\|\mathbf{a} \times \mathbf{b}\|}{\mathbf{a} \cdot \mathbf{b}} = \frac{3}{\sqrt{3}} = \tan\theta.\]
5Step 5: Calculate the Angle
Solve for \(\theta\) using the tangent value obtained: \(\tan\theta = \sqrt{3}\). This corresponds to an angle \(\theta = \frac{\pi}{3}\) radians or 60 degrees.
Key Concepts
Dot ProductCross ProductAngle Between Vectors
Dot Product
The dot product is a fundamental operation in vector calculus. It helps determine how much of one vector goes in the direction of another. For two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the dot product is calculated as \( \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta \), where \( \theta \) is the angle between the vectors.
This formula reveals a crucial relationship: the dot product depends on the cosine of the angle between the vectors. Thus, when the vectors point in the same direction, \( \cos\theta = 1 \), and the dot product equals the product of their magnitudes.
When vectors are perpendicular, \( \cos\theta = 0 \), making the dot product zero. This makes the dot product a helpful tool for identifying orthogonal vectors.
This formula reveals a crucial relationship: the dot product depends on the cosine of the angle between the vectors. Thus, when the vectors point in the same direction, \( \cos\theta = 1 \), and the dot product equals the product of their magnitudes.
When vectors are perpendicular, \( \cos\theta = 0 \), making the dot product zero. This makes the dot product a helpful tool for identifying orthogonal vectors.
Cross Product
In contrast to the dot product, the cross product of vectors \( \mathbf{a} \) and \( \mathbf{b} \) results in a vector rather than a scalar. This vector is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). The magnitude of the cross product is given by \( \|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin\theta \).
Essentially, the cross product measures how perpendicular one vector is to another. If vectors are parallel, then \( \sin\theta = 0 \), resulting in a cross product with zero magnitude.
For the provided exercise, the cross product vector \( \langle 1, 2, 2 \rangle \) indicates direction, while its magnitude \( 3 \) helps determine the vector's size and the angle between the original vectors.
Essentially, the cross product measures how perpendicular one vector is to another. If vectors are parallel, then \( \sin\theta = 0 \), resulting in a cross product with zero magnitude.
For the provided exercise, the cross product vector \( \langle 1, 2, 2 \rangle \) indicates direction, while its magnitude \( 3 \) helps determine the vector's size and the angle between the original vectors.
Angle Between Vectors
Finding the angle between vectors involves using both the dot and cross products. The tangent of the angle \( \theta \) can be calculated using the relationship \( \tan\theta = \frac{\|\mathbf{a} \times \mathbf{b}\|}{\mathbf{a} \cdot \mathbf{b}} \). This ratio ties together the products methods of evaluating the angle.
For instance, in the problem, \( \tan\theta = \frac{3}{\sqrt{3}} = \sqrt{3} \), a common trigonometric value. Thus, the angle \( \theta \) is \( \frac{\pi}{3} \) radians or 60 degrees.
Understanding this process shows how both the dot product and the cross product contribute to geometric interpretations, such as determining the directionality and spatial relations of vectors.
For instance, in the problem, \( \tan\theta = \frac{3}{\sqrt{3}} = \sqrt{3} \), a common trigonometric value. Thus, the angle \( \theta \) is \( \frac{\pi}{3} \) radians or 60 degrees.
Understanding this process shows how both the dot product and the cross product contribute to geometric interpretations, such as determining the directionality and spatial relations of vectors.
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