Problem 53 - Precalculus with Limits

Question

In Exercises 53-58, determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, parallel, or neither. \(\mathbf{u} = \langle -12, 30 \rangle\) \(\mathbf{v} = \langle \frac{1}{2}, -\frac{5}{4} \rangle\)

Step-by-Step Solution

Answer

The vectors \(\langle -12, 30 \rangle\) and \(\langle \frac{1}{2}, -\frac{5}{4} \rangle\) are neither orthogonal nor parallel.

Step 1: Calculate the dot product

We start by calculating the dot product between the vectors \(\mathbf{u}\) and \(\mathbf{v}\), which is defined as \[ \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \] For \(\mathbf{u} = \langle -12, 30 \rangle\) and \(\mathbf{v} = \langle \frac{1}{2}, -\frac{5}{4} \rangle\), the dot product is \[ \mathbf{u} \cdot \mathbf{v}= (-12) \cdot (\frac{1}{2}) + 30 \cdot (-\frac{5}{4}) = -6 - 37.5 = -43.5 \]

Step 2: Examine the result

Since the dot product is not equal to zero (-43.5 ≠ 0), the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are not orthogonal.

Step 3: Check if vectors are parallel

Two vectors are parallel if one is a scalar multiple of the other. We examine whether \(\mathbf{v}\) can be expressed as a scalar multiple of \(\mathbf{u}\). Dividing each element of \(\mathbf{v}\) by the corresponding element of \(\mathbf{u}\), we get: \[ \frac{\frac{1}{2}}{-12} \neq \frac{-\frac{5}{4}}{30} \] Since our division did not yield the same scalar value, the vectors are not parallel.

Step 4: Conclusion

Based on the findings in steps 2 and 3, we can conclude that the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are neither orthogonal nor parallel.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation involving two vectors. It takes two vectors and returns a single number (a scalar). The dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) is calculated using the formula:
\[\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2\]
This result is significant because it carries a lot of information about the relationship between the two vectors. If the dot product is zero, it indicates that the vectors are orthogonal, meaning they are at a 90-degree angle to each other. A positive dot product implies that the vectors point in a generally similar direction, while a negative dot product shows they point in opposite directions. In the given exercise, after calculating the dot product as -43.5, we can determine that the vectors are not orthogonal.

Orthogonal Vectors

Orthogonal vectors are vectors that meet at right angles (90 degrees) to each other. In two-dimensional or three-dimensional space, this concept is similar to 'perpendicular'. The most crucial characteristic of orthogonal vectors is that their dot product is equal to zero:
\[\mathbf{u} \cdot \mathbf{v} = 0\]
This zero result emerges because the cosine of a 90-degree angle is zero, and the dot product includes this cosine in its calculation. In our example, we have already established that \(\mathbf{u}\) and \(\mathbf{v}\) are not orthogonal, since their dot product is not zero. This is a key check when trying to understand vector relationships.

Parallel Vectors

Parallel vectors are vectors that are aligned in the same or exactly opposite direction, irrespective of their magnitude. For vectors in two dimensions, as in our exercise, the concept of being parallel means that one vector is a scalar multiple of the other:
\[\mathbf{u} = k\mathbf{v}\]
where \(k\) is a scalar. When we attempt to find \(k\) by dividing component-wise and get two different results, we conclude that the vectors are not parallel. In the provided exercise, after dividing \(\mathbf{v}\) by \(\mathbf{u}\), we found different scalars, indicating \(\mathbf{u}\) and \(\mathbf{v}\) aren't parallel.

Vector Operations

Working with vectors involves various operations such as addition, subtraction, scalar multiplication, and finding the magnitude. These operations allow us to manipulate and understand vectors in terms of direction and length. For instance, vector addition combines the magnitude and direction of two vectors, while scalar multiplication changes the magnitude of a vector without altering its direction.

In the context of the given exercise, understanding how to calculate the dot product and determining the conditions for vectors being parallel are crucial vector operations that guide us in analyzing the vector relationships. Mastering these operations is essential for any student delving into vector analysis.