Problem 55 - Precalculus with Limits

Question

In Exercises 53-58, determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, parallel, or neither. \(\mathbf{u} = \frac{1}{4}(3\mathbf{i} - \mathbf{j})\) \(\mathbf{v} = 5\mathbf{i} + 6\mathbf{j}\)

Step-by-Step Solution

Answer

To determine this, the dot product of \(\mathbf{u}\) and \(\mathbf{v}\) must be calculated, and then checked if it is equal to zero or if one vector is a scalar multiple of the other. The relation is given by the results of these operations.

Step 1: Calculate the Dot Product

The dot product of \(\mathbf{u}\) and \(\mathbf{v}\) can be found by multiplying the corresponding components of each vector and adding the products together. Apply this operation for \( u = \frac{1}{4}(3i - j) \) and \( v = 5i + 6j \).

Step 2: Check if vectors are Orthogonal

If the dot product calculated in step 1 equals zero, the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal.

Step 3: Check if vectors are Parallel

If the vectors are not orthogonal, check whether they are parallel. Two vectors are parallel if one is a scalar multiple of the other. If \(\mathbf{v}\) is a scalar multiple of \(\mathbf{u}\), they are parallel, otherwise, they are neither orthogonal nor parallel.

Key Concepts

Orthogonal Vectors

Orthogonal vectors are two vectors that do not share any directionality; they meet at a right angle. In mathematics, this is represented when the dot product of the two vectors equals zero. The dot product is a valuable tool as it helps to determine this relationship efficiently. For example, consider the vectors \( \mathbf{u} = \frac{1}{4}(3\mathbf{i} - \mathbf{j}) \) and \( \mathbf{v} = 5\mathbf{i} + 6\mathbf{j} \). To find out if these vectors are orthogonal, calculate the dot product:

• Multiply the \(i\) components: \( \frac{1}{4} \times 3 \times 5 \)
• Multiply the \(j\) components: \(-\frac{1}{4} \times 6 \)
• Add the results: \( \frac{15}{4} - \frac{6}{4} \)

This calculation results in a non-zero value, which means that these vectors are not orthogonal.

Dot Product

The dot product, also known as the scalar product, is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This number is a measure of how much one vector goes in the direction of another. For vectors \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}\) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j}\), the dot product is calculated as:

• \(a_1 \times b_1 + a_2 \times b_2\)

In simpler terms, multiply corresponding components and add them up. If the result is zero, this means the vectors are orthogonal, indicating they intersect at a right angle. For our specific exercise, the vectors yielded a non-zero dot product, which confirms they are not orthogonal.

Parallel Vectors

Parallel vectors, on the other hand, share the same or exact opposite direction. This means one vector can be expressed as a scalar multiple of the other. For example, the vector \( \mathbf{a} \) is parallel to \( \mathbf{b} \) if there exists some scalar \( k \) such that \( \mathbf{a} = k\mathbf{b} \). To determine if two vectors are parallel, compare their components proportionally.

• For \( \mathbf{u} = \frac{1}{4}(3\mathbf{i} - \mathbf{j}) \) and \( \mathbf{v} = 5\mathbf{i} + 6\mathbf{j}\), compute: \(\frac{1}{4} \times 3 / 5\) vs. \(-\frac{1}{4} / 6\)

Here, the respective ratios do not match, as they must for the vectors to be parallel. Hence, these vectors are neither orthogonal nor parallel. Understanding these distinctions can significantly enhance your grasp of vector mathematics by helping you visualize these relationships better in geometric terms.