What's the angle between two vectors if their dot product is equal to the magnitude of their cross product?

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It takes two vectors and returns a single number (scalar). To calculate the dot product of two vectors, you multiply their corresponding components and then add those products together. Mathematically, it's expressed as:
\[ \vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z \]
where \( \vec{a} \) and \( \vec{b} \) are vectors, and \( a_x \), \( a_y \), \( a_z \), \( b_x \), \( b_y \), and \( b_z \) are the components of these vectors in the x, y, and z dimensions, respectively.

Interestingly, the dot product is also related to the cosine of the angle between the two vectors. This relationship is expressed by the formula:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| \cdot |\vec{b}| \cdot \cos(\theta) \]
where \( |\vec{a}| \) and \( |\vec{b}| \) are the magnitudes of the vectors, and \( \theta \) is the angle between them. This property helps determine the angle between vectors and is at the heart of the exercise problem.

In contrast to the dot product, the cross product returns a vector that is perpendicular to both of the original vectors. It is most commonly used in three-dimensional space. The magnitude of the resulting vector is equal to the area of the parallelogram that the two vectors span.
The formula for the cross product is:
\[ \vec{a} \times \vec{b} = (a_y b_z - a_z b_y) \vec{i} + (a_z b_x - a_x b_z) \vec{j} + (a_x b_y - a_y b_x) \vec{k} \]
where \( \vec{i} \), \( \vec{j} \), and \( \vec{k} \) are the unit vectors in the x, y, and z directions, respectively.

The magnitude of the cross product can also be calculated using the sine of the angle between the two vectors:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| \cdot |\vec{b}| \cdot \sin(\theta) \]
This relationship between the cross product's magnitude and the sine function is crucial for understanding when the magnitude of the cross product equals the dot product in magnitude.

The magnitude of a vector reflects its length or size and is often denoted by placing absolute value bars around the vector symbol, like \( |\vec{a}| \). To compute the magnitude of a vector in a two or three-dimensional space, you use the Pythagorean theorem:
\[ |\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2} \]
Where \( a_x \), \( a_y \), and \( a_z \) are the vector's components.

It is important to note that the magnitude of a vector is always non-negative and gives insights into the vector's 'effect' in terms of its contribution to the dot and cross products. A vector with a greater magnitude will have a more significant impact on the results of these operations.

Calculating the angle between two vectors is crucial for understanding their directional relationship. As we've seen with the dot and cross products, these calculations involve trigonometric functions such as sine and cosine.

To find the angle \( \theta \) between two vectors \( \vec{a} \) and \( \vec{b} \), you can use the dot product equation rearranged to solve for the angle:
\[ \cos(\theta) = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \cdot |\vec{b}|} \]
Alternatively, if you have the magnitudes of both vectors and the magnitude of their cross product, you can use the sine function instead:
\[ \sin(\theta) = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}| \cdot |\vec{b}|} \]

In the context of the given exercise, the equality of the magnitudes of the dot and cross products leads to \( \cos(\theta) \) being equal to \( \sin(\theta) \), which only occurs at an angle of 45° or \( \frac{\pi}{4} \) radians, revealing the angle between the two vectors.