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 \(\vec{u} \times \vec{v}\). \(\|\vec{u}\|=3, \quad\|\vec{v}\|=4, \quad \theta=\pi\)

Short answer

The magnitude of \(\vec{u} \times \vec{v}\) is 0.

Step-by-step solution

Determine the Formula

The magnitude of the cross product of two vectors \( \vec{u} \) and \( \vec{v} \) can be found using the formula: \[ \| \vec{u} \times \vec{v} \| = \| \vec{u} \| \cdot \| \vec{v} \| \cdot \sin(\theta). \] This formula requires the magnitudes of vectors and the sine of the angle between them.

Substitute Values

We are given \( \| \vec{u} \| = 3 \), \( \| \vec{v} \| = 4 \), and \( \theta = \pi \). Substitute these values into the formula:\[ \| \vec{u} \times \vec{v} \| = 3 \cdot 4 \cdot \sin(\pi). \]

Calculate the Sine of the Angle

Compute \( \sin(\pi) \). Recall that sine of \( \pi \) is 0. Thus,\[ \sin(\pi) = 0. \]

Compute the Magnitude of the Cross Product

Substitute \( \sin(\pi) = 0 \) back into the equation:\[ \| \vec{u} \times \vec{v} \| = 3 \cdot 4 \cdot 0 = 0. \]Therefore, the magnitude of the cross product \( \| \vec{u} \times \vec{v} \| \) is 0.

Key concepts

Magnitude of Vectors

The magnitude of a vector is a measure of its length. When dealing with vectors in mathematics, especially in three-dimensional space, we often refer to the magnitude, which is denoted as \( \| \vec{u} \| \). For a vector \( \vec{u} = (x, y, z) \) in \( \mathbb{R}^3 \), its magnitude can be calculated using the formula \(\|\vec{u}\| = \sqrt{x^2 + y^2 + z^2}\).
This formula is derived from the Pythagorean theorem and gives us the Euclidean distance of the vector from the origin to the point defined by the vector's components.

• For example, if \(\vec{u}\) has components \(3, 0, 0\), its magnitude is \( \sqrt{3^2 + 0^2 + 0^2} = 3\).

Understanding the magnitude is crucial when calculating other properties of vectors, such as their directions or their cross products.

Angle Between Vectors

The angle between two vectors \(\vec{u}\) and \(\vec{v}\) in \(\mathbb{R}^3\) provides important information about their orientation relative to one another.
The angle is typically denoted as \(\theta\).
To compute the angle, you often use the dot product formula: \(\vec{u} \cdot \vec{v} = \|\vec{u}\|\|\vec{v}\|\cos(\theta)\).
This can be rearranged to solve for the cosine of the angle:

• \(\cos(\theta) = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\|\|\vec{v}\|} \)

While the cosine function often finds the angle, the cross product of vectors, related to the sine function, is useful in different contexts, especially when dealing with perpendicular vectors.
The angle \(\theta\) is essential to determine how 'aligned' or 'orthogonal' two vectors are, which affects their cross product magnitude.

Sine Function

The sine function is a fundamental trigonometric function that comes into play when calculating the magnitude of the cross product of vectors. The value of sine varies based on the angle. For instance:

• \(\sin(0) = 0\)
• \(\sin(\pi/2) = 1\)
• \(\sin(\pi) = 0\)

In vector mathematics, especially in three-dimensional space, the sine of the angle \(\theta\) between two vectors is pivotal in determining their cross product magnitude: \(\| \vec{u} \times \vec{v} \| = \| \vec{u} \| \cdot \| \vec{v} \| \cdot \sin(\theta)\).
This property illustrates how the cross product is maximal when vectors are perpendicular (\(\sin(\pi/2) = 1\)) and zero when vectors are parallel or pointing in antithetic directions (\(\sin(\pi) = 0\)).
Understanding this helps predict the relationship and interaction between two vectors in space.

Vectors in R^3

Vectors in \(\mathbb{R}^3\) denote a direction and magnitude within three-dimensional space. They are expressed as \(\vec{v} = (x, y, z)\) where \(x, y, z\) are the vector's components along each axis.
These vectors are crucial in a vast range of applications, from physics to computer graphics. In 3D space:

• Each vector has a specified length or magnitude.
• The direction of the vector is from the origin to the point \(x, y, z\).

Using the vector cross product is a unique feature of 3D space, as it results in another vector that is perpendicular to the plane containing the initial two vectors.
This is unlike situations in \(\mathbb{R}^2\), where vectors lie on a plane.
Thus, the richness of \(\mathbb{R}^3\) introduces new vector operations and applications that are vital for understanding spatial relations and physical simulations.