Question

In Exercises \(15-20\), determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal parallel, or neither. $$ \begin{array}{l} \mathbf{u}=\langle 4,3\rangle \\ \mathbf{v}=\left\langle\frac{1}{2},-\frac{2}{3}\right\rangle \end{array} $$

Short answer

\(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal.

Step-by-step solution

Calculate the Dot Product

First, compute the dot product of \(\mathbf{u}\) and \(\mathbf{v}\). The dot product is calculated as follows: \((4 * \frac{1}{2}) + (3 * -\frac{2}{3}) = 2 - 2 = 0\)

Verify Parallelity

The vectors are parallel if one vector is a scalar multiple of the other. Divide the components of \(\mathbf{u}\) by the components of \(\mathbf{v}\) to check if they result in the same scalar: \(4/(\frac{1}{2}) =8\) and \(3/(-\frac{2}{3}) = -4.5\). Since these results are not the same means the vectors are not parallel.

Conclusion

Since the dot product is zero and the vectors are not scalar multiples of each other, it can be concluded that the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal.

Key concepts

Dot Product

The dot product, also known as the scalar product, is a crucial operation in vector calculus that serves as a foundation for understanding geometric concepts such as the angle between two vectors. It is given by multiplying corresponding components of two vectors and then summing the products. Mathematically, if two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), their dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as \( a_1 \times b_1 + a_2 \times b_2 \).

In our exercise, the dot product of vectors \( \mathbf{u} \) and \( \mathbf{v} \) is computed as \( (4 \times \frac{1}{2}) + (3 \times -\frac{2}{3}) \), simplifying to \( 2 - 2 = 0 \). The result of the dot product being zero holds special significance; this leads us to a connection with orthogonal vectors, which we will explore in the next section.

Orthogonal Vectors

Orthogonal vectors are those that, in a two-dimensional or three-dimensional space, meet at a right angle (90 degrees) to each other. The concept of orthogonality in vector calculus is identified through the dot product; when the dot product of two vectors is zero, the vectors are orthogonal.

This property is not just a mathematical abstraction but corresponds to the intuitive idea of perpendicularity. For example, the standard axes in the Cartesian coordinate system, \( \mathbf{i} = \langle 1, 0 \rangle \) and \( \mathbf{j} = \langle 0, 1 \rangle \) are orthogonal because \( \mathbf{i} \cdot \mathbf{j} = 0 \). In our given exercise, since the computed dot product is zero, this confirms that vectors \( \mathbf{u} \) and \( \mathbf{v} \) are indeed orthogonal.

Parallel Vectors

Parallel vectors have the same or exactly opposite direction and can be considered as moving along the same line in space, even if they are of different lengths. Mathematical characterization of parallel vectors is made by checking if one vector is a scalar multiple of the other.

For instance, if vector \( \mathbf{p} \) is \( k \) times vector \( \mathbf{q} \), they are considered parallel, which can be denoted as \( \mathbf{p} = k \mathbf{q} \). Contrastingly, the vectors in our exercise are not scalar multiples of each other, as seen by comparing the ratios \( \frac{4}{\frac{1}{2}} \) to \( \frac{3}{-\frac{2}{3}} \) which yields 8 and -4.5 respectively. This discrepancy between the ratios implies that the vectors \( \mathbf{u} \) and \( \mathbf{v} \) are not parallel, supporting our earlier conclusion that they are orthogonal.