Question

Let \(\mathbf{u}\) and \(\mathbf{v}\) be nonzero, nonorthogonal vectors. If \(\theta\) is the angle between them, show that \(\tan \theta=\frac{\|\mathbf{u} \times \mathbf{v}\|}{\mathbf{u} \cdot \mathbf{v}}\).

Short answer

\(\tan \theta = \frac{\|\mathbf{u} \times \mathbf{v}\|}{\mathbf{u} \cdot \mathbf{v}}\) by using properties of the cross and dot products related to sine and cosine.

Step-by-step solution

Understanding the Dot Product and Angle

The dot product between two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by \(\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos\theta\), where \(\theta\) is the angle between them.

Understanding the Cross Product and Angle

The magnitude of the cross product of two vectors is given by \(\|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\| \|\mathbf{v}\| \sin\theta\). This formula relates the magnitude of the cross product to the sine of the angle between the vectors.

Expressing Tangent in terms of Sine and Cosine

Recall that \(\tan\theta = \frac{\sin\theta}{\cos\theta}\). The formula for tangent can be derived by taking the ratio of the sine and cosine of the angle \(\theta\).

Substitute and Simplify

Using the expressions from the previous steps, \(\sin\theta\) can be written as \(\frac{\|\mathbf{u} \times \mathbf{v}\|}{\|\mathbf{u}\|\|\mathbf{v}\|}\) and \(\cos\theta\) as \(\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}\). Substituting these into the tangent formula gives:\[\tan\theta = \frac{\frac{\|\mathbf{u} \times \mathbf{v}\|}{\|\mathbf{u}\|\|\mathbf{v}\|}}{\frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}} = \frac{\|\mathbf{u} \times \mathbf{v}\|}{\mathbf{u} \cdot \mathbf{v}}\]Thus, \(\tan\theta = \frac{\|\mathbf{u} \times \mathbf{v}\|}{\mathbf{u} \cdot \mathbf{v}}\).

Key concepts

Dot Product

The dot product is a crucial operation in vector mathematics. It takes two vectors, say \( \mathbf{u} \) and \( \mathbf{v} \), and returns a scalar. Mathematically, this is expressed as \( \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos\theta \), where \( \|\mathbf{u}\| \) and \( \|\mathbf{v}\| \) are the magnitudes (or lengths) of the vectors, and \( \theta \) is the angle between them.
This formula highlights an important connection: the dot product gives insight into how much one vector extends in the direction of another. If the dot product is positive, the vectors are pointing in more or less the same direction. If it's negative, they point in opposite directions, and if it's zero, the vectors are perpendicular or orthogonal. These properties make the dot product invaluable in both physics and engineering.
Another common use for the dot product is in finding angles. Since it directly involves \( \cos\theta \), we can rearrange the formula to find \( \theta \) if we know the dot product and the magnitudes of the vectors. This ability to relate angle to directionality is what makes the dot product so handy in vector analysis.

Cross Product

The cross product is another fundamental operation in vector mathematics, but unlike the dot product, it returns a vector instead of a scalar. Given two 3-dimensional vectors \( \mathbf{u} \) and \( \mathbf{v} \), the cross product \( \mathbf{u} \times \mathbf{v} \) is a vector that is perpendicular (orthogonal) to both \( \mathbf{u} \) and \( \mathbf{v} \).
The magnitude of this cross product is given by \( \|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\| \|\mathbf{v}\| \sin\theta \). Here, \( \sin\theta \) represents the sine of the angle between the two original vectors. This implies that \( \|\mathbf{u} \times \mathbf{v}\| \) is maximized when the vectors are perpendicular to each other.
The cross product is used in scenarios requiring the notion of rotation or orientation. For example, it's common in physics for determining the torque of a force or the rotational effect in three-dimensional space. The direction of the resulting vector from the cross product can be determined by the right-hand rule, a critical concept when dealing with rotational systems.

Angle Between Vectors

Understanding the angle between vectors is vital for analyzing both their direction and their relation to one another. The angle \( \theta \) can be directly calculated using both the dot and cross products of the vectors, encapsulating their geometric relationships.
By using the formula \( \tan\theta = \frac{\|\mathbf{u} \times \mathbf{v}\|}{\mathbf{u} \cdot \mathbf{v}} \), we express the tangent of the angle using both operations. Here's the breakdown:

• The dot product \( \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\|\|\mathbf{v}\| \cos\theta \) relates to \( \cos\theta \).
• The cross product magnitude \( \|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\|\|\mathbf{v}\| \sin\theta \) corresponds to \( \sin\theta \).

Thus, dividing \( \sin\theta \) by \( \cos\theta \) as per the tangent definition gives \( \tan\theta = \frac{\sin\theta}{\cos\theta} \). Simplifying this with vector products helps to find \( \tan\theta \) in practical applications, providing an elegant method to determine relative angles in space.