Problem 28 - Advanced Engineering Mathematics

Question

Let \(\mathscr{L}_{a}\) and \(\mathscr{L}_{b}\) be lines with direction vectors a and \(\mathbf{b}\), respectively. \(\mathscr{L}_{a}\) and \(\mathscr{L}_{b}\) areorthogonalifa and \(\mathbf{b}\) are orthogonal and parallel if a and \(\mathbf{b}\) are parallel. Determine which of the following lines are orthogonal and which are parallel. (a) \(\mathbf{r}=\langle 1,0,2\rangle+t\langle 9,-12,6\rangle\) (b) \(x=1+9 t, y=12 t, z=2-6 t\) (c) \(x=2 t, y=-3 t, z=4 t\) (d) \(x=5+t, y=4 t, z=3+\frac{5}{2} t\) (e) \(x=1+t, y=\frac{3}{2} t, z=2-\frac{3}{2} t\) (f) \(\frac{x+1}{-3}=\frac{y+6}{4}=\frac{z-3}{-2}\)

Step-by-Step Solution

Answer

None of the lines are orthogonal or parallel.

Step 1: Identify Direction Vectors

First, we need to identify the direction vectors for each line. The direction vector of a line in parametric form can be directly derived from the coefficients accompanying the parameter, usually denoted as \( t \). (a) Direction vector \( \mathbf{a} = \langle 9, -12, 6 \rangle \)(b) Direction vector \( \mathbf{b} = \langle 9, 12, -6 \rangle \)(c) Direction vector \( \mathbf{c} = \langle 2, -3, 4 \rangle \)(d) Direction vector \( \mathbf{d} = \langle 1, 4, \frac{5}{2} \rangle \)(e) Direction vector \( \mathbf{e} = \langle 1, \frac{3}{2}, -\frac{3}{2} \rangle \)(f) Direction vector \( \mathbf{f} = \langle -3, 4, -2 \rangle \).

Step 2: Check Orthogonality

Two vectors are orthogonal if their dot product is zero. We calculate the dot products of all pairs:- \( \mathbf{a} \cdot \mathbf{b} = (9)(9) + (-12)(12) + (6)(-6) = 81 + (-144) + (-36) = -99 \)- \( \mathbf{a} \cdot \mathbf{c} = (9)(2) + (-12)(-3) + (6)(4) = 18 + 36 + 24 = 78 \)- \( \mathbf{a} \cdot \mathbf{d} = (9)(1) + (-12)(4) + (6)\cdot\frac{5}{2} = 9 - 48 + 15 = -24 \)- \( \mathbf{a} \cdot \mathbf{e} = (9)(1) + (-12)(\frac{3}{2}) + (6)(-\frac{3}{2}) = 9 - 18 - 9 = -18 \)- \( \mathbf{a} \cdot \mathbf{f} = (9)(-3) + (-12)(4) + (6)(-2) = -27 - 48 - 12 = -87 \) No pairs are orthogonal as no dot product equals zero.

Step 3: Check Parallelism

Two vectors are parallel if one is a scalar multiple of the other. Let's check by comparing ratios among corresponding elements:- \( \mathbf{a} = \langle 9, -12, 6 \rangle \) and \( \mathbf{b} = \langle 9, 12, -6 \rangle \). Comparing components, 9:9, -12:12, 6:-6 are not consistent ratios.- Check if \( \mathbf{a}, \mathbf{c}, \mathbf{d}, \mathbf{e}, \mathbf{f} \). None are scalar multiples of each other based on component ratios.No pairs of vectors have consistent ratios, so no lines are parallel.

Step 4: Conclusion: Determine Relationships

Since none of the lines have a direction vector that is the scalar multiple of another, they are not parallel. Similarly, since no dot product is zero, no lines are orthogonal. Each line stands independently with no direct parallel or orthogonal relationships as defined here.

Key Concepts

Direction Vectors

Direction vectors are crucial in understanding the orientation of a line in space. For any line given in parametric form, the direction vector can be directly obtained from the coefficients accompanying the parameter \( t \). This vector essentially provides the 'direction' in which the line extends.
For example, if you have a line given by \( \mathbf{r} = \langle x_0, y_0, z_0 \rangle + t \langle a, b, c \rangle \), the direction vector is \( \langle a, b, c \rangle \). Understanding direction vectors can help you to:

• Visualize how a line moves along the axes.
• Determine the orientation of a line relative to others.
• Perform calculations like finding if lines are orthogonal or parallel.

By identifying direction vectors for lines, as shown in the given problem, you form the basis for further operations like calculating the dot product to check for orthogonality.

Orthogonality

Orthogonality between vectors relates to them being at right angles to each other in space. Two vectors are considered orthogonal if their dot product equals zero.

Dot Product and Orthogonality

The dot product of two vectors \( \mathbf{a} \cdot \mathbf{b} \) is calculated as:\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]For vectors to be orthogonal:

• Calculate the dot product.
• If the result is zero, the vectors are orthogonal.

In the context of the exercise, none of the pairs of direction vectors were orthogonal since their dot products were all non-zero. This indicates that none of the lines form a right angle with one another.

Parallelism

Parallelism between vectors means they lie along the same line or planes without intersecting, besides stretching infinitely. Two vectors are parallel if one is a scalar multiple of the other.For two vectors \( \mathbf{a} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{b} = \langle b_1, b_2, b_3 \rangle \) to be parallel, there should exist a constant \( k \) such that:\[ a_1 = kb_1, \quad a_2 = kb_2, \quad a_3 = kb_3 \]

Identifying Parallel Vectors

• Compare the ratios of the corresponding components of the vectors.
• If you find a consistent ratio for each component, the vectors are parallel.

In the exercise, the direction vectors did not exhibit consistent scalar multiplies, meaning the lines are not parallel to each other.

Dot Product

The dot product is an important operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This number can tell us many important things about the vectors, such as their lengths and the angle between them.The dot product \( \mathbf{a} \cdot \mathbf{b} \) is given by the formula:\[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \]Here are a few key insights:

• If \( \mathbf{a} \cdot \mathbf{b} = 0 \), the vectors are orthogonal.
• The dot product helps in calculating projections and understanding vector projections in physics and engineering.

In solving exercises like given, the dot product is a fundamental tool to analyze vector relationships beyond just orthogonality, providing insight into their geometric relationships.