Problem 19 - Essential Calculus Early Transcendentals

Question

\(19-20\) Determine whether the given vectors are orthogonal, parallel, or neither. $$ \begin{array}{l}{\text { (a) } \mathbf{a}=\langle- 5,3,7\rangle, \quad \mathbf{b}=\langle 6,-8,2\rangle} \\ {\text { (b) } \mathbf{a}=\langle 4,6\rangle, \quad \mathbf{b}=\langle- 3,2\rangle} \\ {\text { (c) } \mathbf{a}=-\mathbf{i}+ 2 \mathbf{j}+5 \mathbf{k}, \quad \mathbf{b}=3 \mathbf{i}+4 \mathbf{j}-\mathbf{k}} \\ {\text { (d) } \mathbf{a}=2 \mathbf{i}+6 \mathbf{j}-4 \mathbf{k}, \quad \mathbf{b}=-3 \mathbf{i}-9 \mathbf{j}+6 \mathbf{k}}\end{array} $$

Step-by-Step Solution

Answer

(a) Neither; (b) Orthogonal; (c) Orthogonal; (d) Parallel.

Step 1: Calculate the dot product for (a)

The dot product of two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) is given by \( a_1b_1 + a_2b_2 + a_3b_3 \). For (a), \( \mathbf{a} = \langle -5, 3, 7 \rangle \) and \( \mathbf{b} = \langle 6, -8, 2 \rangle \), calculate the dot product: \(-5 \times 6 + 3 \times (-8) + 7 \times 2 = -30 - 24 + 14 = -40 \). Since the dot product is not zero, the vectors are neither orthogonal nor parallel.

Step 2: Check parallelism for (a)

Vectors are parallel if they are scalar multiples of each other. Check if there exists a constant \( k \) such that \( k \mathbf{a} = \mathbf{b} \) or \( \mathbf{a} = k\mathbf{b} \). For \( \mathbf{a} = \langle -5, 3, 7 \rangle \) and \( \mathbf{b} = \langle 6, -8, 2 \rangle \), ensure that \( \frac{-5}{6} = \frac{3}{-8} = \frac{7}{2} \). These ratios are not equal, thus the vectors are not parallel.

Step 3: Conclusion for (a)

Since the vectors are neither orthogonal (dot product not zero) nor parallel (not scalar multiples), the vectors in part (a) are neither.

Step 4: Calculate the dot product for (b)

For (b), \( \mathbf{a} = \langle 4, 6 \rangle \) and \( \mathbf{b} = \langle -3, 2 \rangle \). The dot product is \( 4 \times (-3) + 6 \times 2 = -12 + 12 = 0 \). A dot product of zero means the vectors are orthogonal.

Step 5: Conclusion for (b)

Since the dot product is zero, the vectors in part (b) are orthogonal.

Step 6: Calculate the dot product for (c)

For (c), \( \mathbf{a} = -\mathbf{i} + 2\mathbf{j} + 5\mathbf{k} \) and \( \mathbf{b} = 3\mathbf{i} + 4\mathbf{j} - \mathbf{k} \). Compute the dot product: \((-1) \times 3 + 2 \times 4 + 5 \times (-1) = -3 + 8 - 5 = 0 \). The dot product is zero, indicating that the vectors are orthogonal.

Step 7: Conclusion for (c)

Since the dot product is zero, the vectors in part (c) are orthogonal.

Step 8: Calculate the dot product for (d)

For (d), \( \mathbf{a} = 2\mathbf{i} + 6\mathbf{j} - 4\mathbf{k} \) and \( \mathbf{b} = -3\mathbf{i} - 9\mathbf{j} + 6\mathbf{k} \). Compute the dot product: \(2 \times (-3) + 6 \times (-9) + (-4) \times 6 = -6 - 54 - 24 = -84 \). The dot product is not zero, indicating the vectors are not orthogonal.

Step 9: Check parallelism for (d)

To check if the vectors are parallel, verify if \( \frac{2}{-3} = \frac{6}{-9} = \frac{-4}{6} \). Simplifying, they all equal \(-\frac{2}{3}\), confirming the vectors are parallel.

Step 10: Conclusion for (d)

Since the vectors are scalar multiples of each other, the vectors in part (d) are parallel.

Key Concepts

Orthogonal Vectors

Orthogonal vectors are vectors that meet at a right angle, i.e., 90 degrees. The main way to determine if two vectors are orthogonal is by calculating their dot product. The dot product, if zero, implies that the vectors are perpendicular to each other.

In practical terms, if you have vectors \( \mathbf{a} \) and \( \mathbf{b} \), and their dot product comes out to be zero, you can confidently say they are orthogonal. Mathematically, the dot product of vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) is \( a_1b_1 + a_2b_2 + a_3b_3 \).

For example, in exercise 19-20, part (b) and (c), the dot product of the vectors is zero, suggesting these vectors are indeed orthogonal. This concept chiefly relies on the geometric understanding that orthogonal vectors, when plotted, would visually appear as having a T-shape or being completely non-aligned.

Parallel Vectors

Parallel vectors are those that point in the same or exactly opposite direction. To decide if two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel, we check if one is a scalar multiple of the other. This means you can multiply every component of one vector by the same non-zero constant to get the other vector.

Imagine you have vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), then \( \mathbf{a} \) is parallel to \( \mathbf{b} \) if there exists a constant \( k \) such that \( k \mathbf{a} = \mathbf{b} \) (or \( \mathbf{a} = k \mathbf{b} \)). The proportions \( \frac{a_1}{b_1}, \frac{a_2}{b_2}, \frac{a_3}{b_3} \) should all be equal to this constant.

In the provided exercise, it’s clearly shown in part (d) that the vectors are parallel because all the calculated ratios are equivalent, confirming they are scaled copies of each other. Parallel vectors may visually overlap perfectly or lie on the same line when plotted.

Dot Product

The dot product is a key calculation in vector analysis that operates on two vectors, resulting in a scalar value. Unlike vector multiplication that typically results in another vector, a dot product answers how much one vector extends in the direction of another.

For vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \), the dot product is computed as \( a_1b_1 + a_2b_2 + a_3b_3 \). It not only helps identify orthogonality, as seen when the dot product is zero, but also has applications in finding angles between vectors and projections.

Throughout the exercise provided, the dot product serves as the initial test for vector relationships, indicating orthogonality when zero and assisting in subsequent checks for parallelism. It is a fundamental element of vector mathematics, setting the stage for deeper insights into vector directions and magnitudes.