Problem 9
Question
In Exercises \(9-14,\) sketch the coordinate axes and then include the vectors \(\mathbf{u}, \mathbf{v}\) and \(\mathbf{u} \times \mathbf{v}\) as vectors starting at the origin. $$ \mathbf{u}=\mathbf{i}, \quad \mathbf{v}=\mathbf{j} $$
Step-by-Step Solution
Verified Answer
\( \mathbf{u} = \mathbf{i} \) along x-axis, \( \mathbf{v} = \mathbf{j} \) along y-axis, \( \mathbf{u} \times \mathbf{v} = \mathbf{k} \) perpendicular to plane, pointing out.
1Step 1: Understand the Vectors
The vectors given are \( \mathbf{u} = \mathbf{i} \) and \( \mathbf{v} = \mathbf{j} \). Vector \( \mathbf{i} \) is the unit vector along the x-axis, and vector \( \mathbf{j} \) is the unit vector along the y-axis.
2Step 2: Sketch the Coordinate Axes
Draw a two-dimensional coordinate plane with a horizontal x-axis and a vertical y-axis. Mark the origin where these two axes intersect.
3Step 3: Plot Vector \( \mathbf{u} \)
Starting from the origin (0,0), draw vector \( \mathbf{u} = \mathbf{i} \) along the positive x-axis towards (1,0).
4Step 4: Plot Vector \( \mathbf{v} \)
Starting from the origin (0,0), draw vector \( \mathbf{v} = \mathbf{j} \) along the positive y-axis towards (0,1).
5Step 5: Compute Cross Product \( \mathbf{u} \times \mathbf{v} \)
The cross product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is given by \( \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 0 & 0 \ 0 & 1 & 0 \end{vmatrix} = \mathbf{k} \). Thus, \( \mathbf{u} \times \mathbf{v} \) is the unit vector along the z-axis.
6Step 6: Visualize Vector \( \mathbf{u} \times \mathbf{v} \)
Since \( \mathbf{u} \times \mathbf{v} = \mathbf{k} \), visualize or if possible, imagine this vector pointing upwards out of the plane, perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \).
Key Concepts
Coordinate SystemUnit VectorsCross ProductThree-Dimensional Space
Coordinate System
A coordinate system allows us to define the position of points or vectors in space. In a two-dimensional system, we usually have two axes: the x-axis, which is horizontal, and the y-axis, which is vertical. These axes intersect at the origin, denoted by the point (0,0). The origin serves as a reference point for determining the position of other points or vectors.
For three-dimensional space, we introduce an additional axis called the z-axis. This axis is typically depicted as coming out of the plane formed by the x and y axes. This setup allows us to locate any point in three-dimensional space using a set of three coordinates like (x, y, z). Each axis helps in determining only one coordinate of this triplet.
In vector mathematics, vectors are often represented in these coordinate systems starting from the origin and pointing towards a particular direction indicated by their components, just like in this exercise with vectors \( \mathbf{u} \) and \( \mathbf{v} \).
For three-dimensional space, we introduce an additional axis called the z-axis. This axis is typically depicted as coming out of the plane formed by the x and y axes. This setup allows us to locate any point in three-dimensional space using a set of three coordinates like (x, y, z). Each axis helps in determining only one coordinate of this triplet.
In vector mathematics, vectors are often represented in these coordinate systems starting from the origin and pointing towards a particular direction indicated by their components, just like in this exercise with vectors \( \mathbf{u} \) and \( \mathbf{v} \).
Unit Vectors
Unit vectors are vectors that have a magnitude of 1. In a coordinate system, they are used to define the direction of the axes. The standard unit vectors in three-dimensional space are \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) which point along the x, y, and z axes respectively.
- \( \mathbf{i} = (1, 0, 0) \) is the unit vector along the x-axis.
- \( \mathbf{j} = (0, 1, 0) \) is the unit vector along the y-axis.
- \( \mathbf{k} = (0, 0, 1) \) is the unit vector along the z-axis.
Using unit vectors makes calculations involving directions straightforward. You can express any vector in terms of these unit vectors and their respective scalar components. For example, the vector \( \mathbf{u} = \mathbf{i} \) indicates it's purely along the x-axis.
- \( \mathbf{i} = (1, 0, 0) \) is the unit vector along the x-axis.
- \( \mathbf{j} = (0, 1, 0) \) is the unit vector along the y-axis.
- \( \mathbf{k} = (0, 0, 1) \) is the unit vector along the z-axis.
Using unit vectors makes calculations involving directions straightforward. You can express any vector in terms of these unit vectors and their respective scalar components. For example, the vector \( \mathbf{u} = \mathbf{i} \) indicates it's purely along the x-axis.
Cross Product
The cross product is an operation on two vectors in three-dimensional space that results in another vector perpendicular to both. It is a vector product, unlike the dot product which results in a scalar.
To compute the cross product of two vectors \( \mathbf{u} = \mathbf{i} \) and \( \mathbf{v} = \mathbf{j} \), we set up a determinant:\[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 0 & 0 \ 0 & 1 & 0 \end{vmatrix} = \mathbf{k} \] This shows that the cross product points in the direction of \( \mathbf{k} \), meaning along the z-axis.
The magnitude of the cross product equals the area of the parallelogram formed by the two vectors. The orientation (direction on the z-axis) is determined by the right-hand rule, where if you point your index finger along \( \mathbf{u} \) and your middle finger along \( \mathbf{v} \), your thumb points in the direction of \( \mathbf{u} \times \mathbf{v} \).
To compute the cross product of two vectors \( \mathbf{u} = \mathbf{i} \) and \( \mathbf{v} = \mathbf{j} \), we set up a determinant:\[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 0 & 0 \ 0 & 1 & 0 \end{vmatrix} = \mathbf{k} \] This shows that the cross product points in the direction of \( \mathbf{k} \), meaning along the z-axis.
The magnitude of the cross product equals the area of the parallelogram formed by the two vectors. The orientation (direction on the z-axis) is determined by the right-hand rule, where if you point your index finger along \( \mathbf{u} \) and your middle finger along \( \mathbf{v} \), your thumb points in the direction of \( \mathbf{u} \times \mathbf{v} \).
Three-Dimensional Space
Three-dimensional space is an extension of our usual two-dimensional perspective in which we consider an additional dimension, represented by the z-axis. This provides a fuller picture by considering length, width, and height as opposed to just length and width in 2D.
In such a space, any point or vector can be described using three coordinates (x, y, z). The addition of the third dimension allows for complex geometries and motion that cannot be captured in a two-dimensional plane.
For vector analysis in three-dimensional spaces, using the z-axis allows us to depict vectors pointing perpendicular to the 2D plane, which is an important concept in problems involving physics and engineering. In our example, after computing the cross product of two vectors \( \mathbf{u} = \mathbf{i} \) and \( \mathbf{v} = \mathbf{j} \), the resulting vector \( \mathbf{u} \times \mathbf{v} = \mathbf{k} \) lies along this z-axis, emphasizing its perpendicularly emergent nature. It showcases how vectors can spread into the depth of three-dimensional space.
In such a space, any point or vector can be described using three coordinates (x, y, z). The addition of the third dimension allows for complex geometries and motion that cannot be captured in a two-dimensional plane.
For vector analysis in three-dimensional spaces, using the z-axis allows us to depict vectors pointing perpendicular to the 2D plane, which is an important concept in problems involving physics and engineering. In our example, after computing the cross product of two vectors \( \mathbf{u} = \mathbf{i} \) and \( \mathbf{v} = \mathbf{j} \), the resulting vector \( \mathbf{u} \times \mathbf{v} = \mathbf{k} \) lies along this z-axis, emphasizing its perpendicularly emergent nature. It showcases how vectors can spread into the depth of three-dimensional space.
Other exercises in this chapter
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