Question

Demonstrate the ambiguity that results when the cross product is used to find the angle between two vectors by finding the angle between \(\mathbf{A}=3 \mathbf{a}_{x}-2 \mathbf{a}_{y}+4 \mathbf{a}_{z}\) and \(\mathbf{B}=2 \mathbf{a}_{x}+\mathbf{a}_{y}-2 \mathbf{a}_{z} .\) Does this ambiguity exist when the dot product is used?

Short answer

Answer: By using the dot product method, we can resolve the ambiguity in finding the angle between two vectors A and B, as the cosine function gives a unique angle within the range of [0, π].

Step-by-step solution

Calculate the cross product of A and B

The cross product of A and B can be found as: \(\mathbf{A} \times \mathbf{B} = ((-2)(-2) - (4)(1))\mathbf{a}_{x} + ((4)(2) - (3)(-2))\mathbf{a}_{y} + ((3)(1) - (2)(-2))\mathbf{a}_{z}\) Calculating the values, we get: \(\mathbf{A} \times \mathbf{B} = 4\mathbf{a}_{x} + 14\mathbf{a}_{y} + 7\mathbf{a}_{z}\)

Calculate the magnitudes of A, B, and AxB

To find the angle, we need the magnitudes of A, B, and AxB: \(|\mathbf{A}| = \sqrt{(3^2) + (-2^2) + (4^2)} = \sqrt{29}\) \(|\mathbf{B}| = \sqrt{(2^2) + (1^2) + (-2^2)} = \sqrt{9} = 3\) \(|\mathbf{A} \times \mathbf{B}| = \sqrt{(4^2) + (14^2) + (7^2)} = \sqrt{249}\)

Find the angle between A and B using cross product

Using the formula for the angle between two vectors using the cross product: \(\sin{\theta} = \frac{|\mathbf{A} \times \mathbf{B}|}{|\mathbf{A}| \cdot |\mathbf{B}|}\) \(\sin{\theta} = \frac{\sqrt{249}}{\sqrt{29} \cdot 3}\) Now, we can see the ambiguity in the cross-product method as the sine function has multiple solutions for an angle, usually leading to two possible angles. Therefore, we need to resolve this ambiguity by finding a unique angle.

Calculate the dot product of A and B

The dot product of A and B is given by, \(\mathbf{A} \cdot \mathbf{B} = (3)(2) + (-2)(1) + (4)(-2) = 6 - 2 - 8 = -4\)

Find the angle between A and B using dot product

Using the formula for the angle between two vectors using the dot product: \(\cos{\theta} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| \cdot |\mathbf{B}|}\) \(\cos{\theta} = \frac{-4}{\sqrt{29} \cdot 3}\) Now, we can clearly see there's no ambiguity with the dot product. The cosine function gives a unique angle as the output, with a range of \([0, \pi]\). So, we can confirm that the ambiguity does not exist when using the dot product to find the angle between two vectors.

Key concepts

Cross Product

The cross product is a vector operation used especially in three-dimensional space. It takes two vectors and produces another vector that is perpendicular to the plane formed by the original vectors. This result is called the normal vector. An important aspect of the cross product is its magnitude, which equals the area of the parallelogram spanned by the two vectors.

• To calculate the cross product of vectors \(\mathbf{A}\) and \(\mathbf{B}\), use the formula: \(\mathbf{A} \times \mathbf{B} = (A_yB_z - A_zB_y)\mathbf{i} + (A_zB_x - A_xB_z)\mathbf{j} + (A_xB_y - A_yB_x)\mathbf{k}\).
• The cross product is not commutative, meaning \(\mathbf{A} \times \mathbf{B} eq \mathbf{B} \times \mathbf{A}\).
• The direction of the vector produced by the cross product follows the right-hand rule.

When using the cross product to find angles between vectors, an ambiguity arises due to the nature of the sine function, which can have multiple solutions. These need to be considered to avoid misinterpretation.

Dot Product

The dot product is another fundamental vector operation that yields a scalar rather than a vector. It is a measure of how much one vector extends in the direction of another and can be used to determine angles, determine projection, and much more.

• The dot product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) is calculated as \(\mathbf{A} \cdot \mathbf{B} = A_xB_x + A_yB_y + A_zB_z\).
• The result is a number, not another vector, and can be positive, negative, or zero.
• A dot product of zero indicates that the vectors are perpendicular (orthogonal).

The angle between two vectors can also be derived using the dot product. This method is often preferred for its lack of ambiguity, since the output of the cosine function for angles is unique within its range of \([0, \pi]\).

Vector Magnitude

Vector magnitude provides a measure of the vector's length. It represents how long the vector is in space, regardless of its direction. For any vector \(\mathbf{A} = (A_x, A_y, A_z)\), its magnitude is given by the formula:

• \(|\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}\)
• Magnitude is always a non-negative value.
• If the magnitude is zero, the vector is called a zero vector, which has no direction.

Understanding vector magnitude is crucial when working with both the cross and dot product, as these operations often require the magnitudes of the vectors involved to complete calculations such as those for the angle between vectors.

Angle Between Vectors

Determining the angle between vectors is essential in analyzing their geometric relationship in space. This angle illustrates how two vectors are oriented concerning each other.

• The angle \(\theta\) between vectors can be found using both cross product and dot product formulations.
• For the cross product, the sine function is used: \(\sin{\theta} = \frac{|\mathbf{A} \times \mathbf{B}|}{|\mathbf{A}| \cdot |\mathbf{B}|}\).
• For the dot product, the cosine function is used: \(\cos{\theta} = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| \cdot |\mathbf{B}|}\).

While the cross product can yield two potential angles due to the properties of sine, the dot product provides a single, clear value, making it effective for resolving ambiguities in angle measurement.