Question

The magnitudes of vectors \(\vec{u}\) and \(\vec{v}\) in \(\mathbb{R}^{3}\) are given, along with the angle \(\theta\) between them. Use this information to find the magnitude of \(\vec{u} \times \vec{v}\). \(\|\vec{u}\|=2, \quad\|\vec{v}\|=5, \quad \theta=5 \pi / 6\)

Short answer

The magnitude of the cross product \(\|\vec{u} \times \vec{v}\|\) is 5.

Step-by-step solution

Understanding Cross Product Magnitude

To find the magnitude of the cross product of two vectors, \(\|\vec{u} \times \vec{v}\|\), we use the formula: \[ \|\vec{u} \times \vec{v}\| = \|\vec{u}\| \cdot \|\vec{v}\| \cdot \sin \theta \] where \(\|\vec{u}\|\) and \(\|\vec{v}\|\) are the magnitudes of the vectors and \(\theta\) is the angle between them.

Substitute Values into Formula

Given the values \(\|\vec{u}\| = 2\), \(\|\vec{v}\| = 5\), and \(\theta = \frac{5\pi}{6}\), substitute these into the formula: \[ \|\vec{u} \times \vec{v}\| = 2 \cdot 5 \cdot \sin \left(\frac{5\pi}{6}\right) \]

Calculate Sine of the Angle

Calculate \(\sin \left(\frac{5\pi}{6}\right)\). The angle \(\frac{5\pi}{6}\) radians corresponds to 150 degrees, and \(\sin(150^\circ) = \sin(30^\circ) = \frac{1}{2}\). Therefore, \(\sin \left(\frac{5\pi}{6}\right) = \frac{1}{2}\).

Compute the Magnitude

Now compute the magnitude: \[ \|\vec{u} \times \vec{v}\| = 2 \cdot 5 \cdot \frac{1}{2} = 5 \] Thus, the magnitude of the cross product is 5.

Key concepts

Cross Product

The cross product of two vectors in three-dimensional space, denoted as \( \vec{u} \times \vec{v} \), results in a vector that is orthogonal (perpendicular) to both original vectors. This product is unique to three-dimensional space and is not applicable in two dimensions. To understand how it works, consider the following:

• The direction of the cross product vector is determined by the right-hand rule: if you point your index finger in the direction of \( \vec{u} \) and your middle finger in the direction of \( \vec{v} \), your thumb will point in the direction of \( \vec{u} \times \vec{v} \).
• The cross product is anti-commutative, which means that \( \vec{u} \times \vec{v} = - (\vec{v} \times \vec{u}) \).
• The magnitude of \( \vec{u} \times \vec{v} \) can be calculated using the formula \( \|\vec{u} \times \vec{v}\| = \|\vec{u}\| \cdot \|\vec{v}\| \cdot \sin \theta \), where \( \theta \) is the angle between \( \vec{u} \) and \( \vec{v} \).

Vector Magnitude

The magnitude of a vector represents its length or size, which is the distance from the origin to the point it represents when placed in Cartesian coordinates. For a vector \( \vec{u} = (u_1, u_2, u_3) \) in three-dimensional space, the magnitude is calculated using the formula:
\[ \|\vec{u}\| = \sqrt{u_1^2 + u_2^2 + u_3^2} \] This formula is derived from the Pythagorean theorem. It provides the length of the vector as if it were the hypotenuse of a right triangle formed by its components. Understanding vector magnitudes is crucial when calculating other vector operations such as the cross product, where the magnitudes of the vectors play a direct role in determining the resultant vector's magnitude.
When comparing vectors, magnitude gives you an idea of which vector has more 'strength' or 'influence'.

Sine Function

The sine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse. In the context of vectors, the sine of the angle between two vectors is used in the formula for the magnitude of the cross product:
\[ \sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}} \] When dealing with vectors, \( \theta \) is the angle between them, and its sine can be obtained easily using a scientific calculator or trigonometric tables. In the step-by-step solution provided earlier, we calculated \( \sin \left(\frac{5\pi}{6}\right) \), which corresponds to an angle of 150 degrees, and found it to be \( \frac{1}{2} \). This value is crucial as it affects the magnitude of the cross product, demonstrating the direct link between the angle (measured in radians or degrees) and the resulting magnitude.

Angle Between Vectors

The angle between two vectors is an important measure that describes how vectors relate to each other in space. This angle \( \theta \) can be calculated if you know the dot product of the vectors, \( \vec{u} \cdot \vec{v} \), and their magnitudes. The formula is:
\[ \cos \theta = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\| \cdot \|\vec{v}\|} \] However, when computing the cross product's magnitude, we use the sine of this angle rather than the cosine. The angle reveals how vectors are aligned or misaligned:

• If \( \theta = 0 \) degrees, vectors are perfectly aligned (parallel).
• If \( \theta = 90 \) degrees, vectors are perpendicular.
• If \( \theta > 0 \) but less than 90 degrees, they form an acute angle.
• If \( 90 < \theta \leq 180 \), they form an obtuse angle.

In the exercise, \( \theta = \frac{5\pi}{6} \) radians, or 150 degrees, forms an obtuse angle, affecting the sine function's value for the cross product magnitude.