Question

Vector \(\overrightarrow{\mathbf{a}}\) has magnitude \(3.20\) units and lies in the \(y z\) plane \(63.0^{\circ}\) from the \(+y\) axis with a positive \(z\) component. Vector \(\overrightarrow{\mathbf{b}}\) has magnitude \(1.40\) units and lies in the \(x z\) plane \(48.0^{\circ}\) from the \(+x\) axis with a positive \(z\) component. Find \(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}\).

Short answer

The cross product of vectors \( \overrightarrow{\mathbf{a}} \) and \( \overrightarrow{\mathbf{b}} \) is equal to \( (3.2\sin(63.0) * 1.4\sin(48.0)- 0) \hat{i} - (0-0) \hat{j} + (0- 3.2\sin(63.0)*1.4\cos(48.0)) \hat{k}\)

Step-by-step solution

Express the vectors in terms of their components

The vector \( \overrightarrow{\mathbf{a}}\) can be expressed in spherical coordinates as: \( \overrightarrow{\mathbf{a}} = 3.20\sin(63.0) \hat{j} + 3.20\cos(63.0) \hat{k}\)\n Likewise, the vector \( \overrightarrow{\mathbf{b}}\) can be expressed as: \( \overrightarrow{\mathbf{b}} = 1.40\cos(48.0)\hat{i} +1.40\sin(48.0)\hat{k} \)

Find the cross product of two vectors

The cross product of any two vectors can be found using the determinant of a 3x3 matrix constructed as follows: \nFirst column: the unit vectors \( \hat{i}, \hat{j}, \hat{k} \)\nSecond column: the components of vector \( \overrightarrow{\mathbf{a}}\) along \( \hat{i}, \hat{j}, \hat{k} \)\n Third column: the components of vector \( \overrightarrow{\mathbf{b}}\) along \( \hat{i}, \hat{j}, \hat{k} \)\n Using these, \(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}\) can be represented as: \(\begin{bmatrix} \hat{i} & \hat{j} & \hat{k} \ 0 & 3.2\sin(63.0) & 3.2\cos(63.0) \ 1.4\cos(48.0) & 0 & 1.4\sin(48.0)\end{bmatrix}\)

Evaluate the Determinant

We can evaluate this expression by following the Sarrus rule for the determinant. Doing this allows us to calculate the \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) components of \(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}\). After performing the calculation, we have:\n \(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}} = (3.2\sin(63.0) * 1.4\sin(48.0)- 0) \hat{i} - (0-0) \hat{j} + (0- 3.2\sin(63.0)*1.4\cos(48.0)) \hat{k}\)

Key concepts

Vector Components

Understanding vector components is essential for analyzing motion, force, and various physical phenomena in different directions. A vector can be broken down into its individual components that indicate its influence in the separate directions of a coordinate system. For instance, in a three-dimensional space, a vector's direction and magnitude are represented by its components along the x, y, and z axes.

To break it down, consider a vector \( \overrightarrow{\mathbf{a}} \) as described in the exercise. Although it's initially given in a particular plane, it still has components that can be projected onto the three principal axes. In our case, even though \( \overrightarrow{\mathbf{a}} \) is in the yz-plane, it doesn't have an x-component; therefore, its influence along the x-axis is zero. Similarly, another vector can have different magnitudes and directions in the xy, yz, or xz planes, distinguishing its unique spatial orientation.

To visualize this, imagine the vector as an arrow in space: its length is the magnitude, and where it points covers the directional components. Each component is like a shadow the vector casts onto the walls of a box - the walls representing the x, y, and z axes. Computing these 'shadows' lets us delve into the specific influence a vector has within a coordinate system, which is pivotal for subsequent operations like the cross product.

Spherical Coordinates

Spherical coordinates offer a three-dimensional coordinate system that extends polar coordinates into three dimensions. It's an alternative to the Cartesian coordinate system and particularly useful for tackling problems featuring symmetry around a point, like those involving orbital motions or electromagnetic fields.

In spherical coordinates, a point in space is determined by three values: the radial distance from the origin (\( r \)), the inclination angle from the z-axis (\( \theta \)), and the azimuthal angle in the xy-plane from the x-axis (\( \phi \)). When we look at vectors in spherical coordinates, these angles and radial distance help in defining the vector's direction and magnitude.

For the given exercise, we see vectors positioned in specific planes with particular angles mentioned. Translating our vector's specifications from the question into Cartesian components using spherical coordinates helps in realizing the complete picture of these vectors’ directions and magnitudes. This conversion—like transcribing text from one language to another—allows for easier manipulation and operation with the vectors, like finding the cross product.

Matrix Determinant

The determinant of a matrix is a special number that can be calculated from its elements. In the context of a 3x3 matrix, the determinant is particularly significant as it can be used to compute the cross product of two vectors. The determinant reflects the 'volume scaling factor' of the linear transformation described by the matrix and can also indicate if a matrix has an inverse, a concept linked to solving systems of linear equations.

For vectors, the matrix structure consists of three rows: the top row is typically occupied by the unit vectors, while the subsequent rows are the components of the vectors in question. The determinant of such a specially constructed matrix yields the resulting cross product’s vector components. As shown in the steps of the provided solution, expanding the determinant by following a rule like Sarrus' rule, or the more common cofactor expansion method, gives us the respective components of the cross product vector, each component multiplying a corresponding unit vector.

Concluding, the matrix determinant isn't just a number but a gateway to understanding the geometric and algebraic characteristics of vector operations in multi-dimensional space—something that underscores the cross product's capability to yield a vector that is orthogonal to the original pair of vectors.