Question

Determine which of the vectors is (are) parallel to \(\mathrm{z}\). Use a graphing utility to confirm your results. \(\mathbf{z}\) has initial point (5,4,1) and terminal point (-2,-4,4) (a) \langle 7,6,2\rangle (b) \langle 14,16,-6\rangle

Short answer

Vector (b) \(\langle 14, 16, -6 \rangle\) is parallel to vector \(\mathbf{z}\) while vector (a) \(\langle 7,8,2 \rangle\) is not.

Step-by-step solution

Compute Vector \(\mathbf{z}\)'s Components

The components of a vector are obtained by subtracting the initial point coordinates from the terminal point coordinates. So, for \(\mathbf{z}\), with initial point (5,4,1) and terminal point (-2,-4,4), the components are (-2-5, -4-4, 4-1) = (-7, -8, 3).

Comparing Vectors

We have vector (a) as \(\langle 7,8,2 \rangle\) and vector (b) as \(\langle 14, 16, -6 \rangle\). Now, vectors are parallel if they are scalar multiples of each other. Divide the components of vector (a) and (b) by the corresponding components of \(\mathbf{z}\). For instance, the x-component of (a) divided by the x-component of \(\mathbf{z}\) is \(7/-7 = -1\). Do this for all components.

Making Conclusions

If the ratios obtained in Step 2 are equal, then the vectors are parallel. For vector (a), we get the ratios as -1, -1, and 2/3. Since these ratios are not the same, vector (a) is not parallel to \(\mathbf{z}\). For vector (b), the ratios are -2, -2, and -2 which are all equal, so vector (b) is parallel to \(\mathbf{z}\).

Graphical Verification

Graph all vectors using a graphing utility. The direction of the vectors is visualized by the direction of the arrows. Parallel vectors will appear as lines that never intersect and are equidistant from each other. Vectors \(\mathbf{z}\) and (b) will have this appearance confirming that they are indeed parallel, whereas vector (a) will not share this characteristic with \(\mathbf{z}\).

Key concepts

Vector Components

Understanding vector components is crucial for analyzing the direction and magnitude of vectors. In the context of this exercise, we begin by identifying the components of vector \(\mathbf{z}\). The components are the changes in the x, y, and z coordinates from the initial to the terminal point. For vector \(\mathbf{z}\), this transition from the initial point \( (5,4,1) \) to the terminal point \( (-2,-4,4) \) results in the vector components \((-7, -8, 3)\).

These components are an essential first step; they provide the directional information that is key when determining parallelism between vectors. It's helpful to visualize this as a directed line segment in space, marked by a change along each of the three axes. In short, the components of a vector are the tools that unlock an understanding of a vector's movement through space.

Scalar Multiples

When we say vectors are scalar multiples of each other, we are essentially referring to a proportional relationship. Two vectors are parallel if one is a scalar multiple of the other. This means you can multiply or stretch one vector by a certain amount, a scalar, to get the other vector.

In the given example, we observe this by comparing the components. For instance, vector (b) has the components \(\langle 14, 16, -6 \rangle\), and when each component is divided by the corresponding ones of vector \(\mathbf{z}\), we get equal ratios of \(-2\). This uniform scaling factor confirms that vector (b) is a scalar multiple of vector \(\mathbf{z}\), and hence they are parallel.

However, vector (a) produces varying ratios, indicating the absence of a consistent scalar multiple, which in turn shows that vector (a) is not parallel to vector \(\mathbf{z}\). Scalar multiplication is a fundamental concept in vector operations that illustrates the idea of vectors being in the same line of action, albeit at different magnitudes.

Graphing Utility

Graphing utilities are invaluable tools in confirming observations made algebraically. They provide a visual representation of how vectors behave in space. When you plot vectors on a graph, parallel vectors appear as lines or arrows that, while possibly being of different lengths, will never meet or intersect no matter how far they are extended.

In our exercise, graphing vector \(\mathbf{z}\) and vector (b) would illustrate lines that run in the exact same direction, reinforcing their parallel nature as concluded algebraically. On the other hand, graphing vector (a) beside \(\mathbf{z}\) will show a different orientation, negating parallelism. This graphical method supports the numerical comparison and serves as a double-check to ensure our understanding of vector relationships is both mathematically sound and visually congruent. It's a powerful demonstration of mathematical concepts in a visual format.