Problem 20 - Essential Calculus Early Transcendentals

Question

\(19-20\) Determine whether the given vectors are orthogonal, parallel, or neither. $$\begin{array}{ll}{\text { (a) } \mathbf{u}=\langle- 3,9,6\rangle,}\quad {\mathbf{v}=\langle 4,-12,-8\rangle} \\ {\text { (b) } \mathbf{u}} {=\mathbf{i}-\mathbf{j}+ 2 \mathbf{k}, \quad \mathbf{v}=2 \mathbf{i}-\mathbf{j}+\mathbf{k}} \\ {(\mathrm{c}) \mathbf{u}} {=\langle a, b, c\rangle, \quad \mathbf{v}=\langle- b, a, 0\rangle}\end{array}$$

Step-by-Step Solution

Answer

(a) Parallel, (b) Neither, (c) Orthogonal.

Step 1: Determine if Vectors (a) are Orthogonal

To determine if the vectors \( \mathbf{u} = \langle -3, 9, 6 \rangle \) and \( \mathbf{v} = \langle 4, -12, -8 \rangle \) are orthogonal, compute their dot product. Two vectors are orthogonal if their dot product is zero.\[\mathbf{u} \cdot \mathbf{v} = (-3)(4) + (9)(-12) + (6)(-8) = -12 - 108 - 48 = -168 \]Since the dot product is not zero, the vectors are not orthogonal.

Step 2: Determine if Vectors (a) are Parallel

Vectors are parallel if they are scalar multiples of each other. To check if \( \mathbf{u} = \langle -3, 9, 6 \rangle \) and \( \mathbf{v} = \langle 4, -12, -8 \rangle \) are parallel, find a scalar \( k \) such that \( \mathbf{v} = k \mathbf{u} \):Checking components, \(\frac{4}{-3} = \frac{-12}{9} = \frac{-8}{6} = -\frac{4}{3}\).Since all components have the same ratio, the vectors are parallel.

Step 3: Determine if Vectors (b) are Orthogonal

To find out if \( \mathbf{u} = \mathbf{i} - \mathbf{j} + 2 \mathbf{k} \) and \( \mathbf{v} = 2 \mathbf{i} - \mathbf{j} + \mathbf{k} \) are orthogonal, calculate their dot product:\[\mathbf{u} \cdot \mathbf{v} = (1)(2) + (-1)(-1) + (2)(1) = 2 + 1 + 2 = 5\]Since the dot product is not zero, the vectors are not orthogonal.

Step 4: Determine if Vectors (b) are Parallel

To check if \( \mathbf{u} = \mathbf{i} - \mathbf{j} + 2 \mathbf{k} \) and \( \mathbf{v} = 2 \mathbf{i} - \mathbf{j} + \mathbf{k} \) are parallel, see if \( \mathbf{v} \) is a scalar multiple of \( \mathbf{u} \):Examining component ratios: \( \frac{2}{1} eq \frac{-1}{-1} eq \frac{1}{2} \) shows different ratios.Therefore, the vectors are neither parallel nor orthogonal.

Step 5: Determine if Vectors (c) are Orthogonal

For vectors \( \mathbf{u} = \langle a, b, c \rangle \) and \( \mathbf{v} = \langle -b, a, 0 \rangle \), compute their dot product to check orthogonality:\[\mathbf{u} \cdot \mathbf{v} = (a)(-b) + (b)(a) + (c)(0) = -ab + ab + 0 = 0\]The dot product is zero, so the vectors are orthogonal.

Step 6: Simplify Result

We have determined in previous steps: - For (a): Vectors are parallel. - For (b): Vectors are neither orthogonal nor parallel. - For (c): Vectors are orthogonal.

Key Concepts

Orthogonal Vectors

Orthogonal vectors are special because they are, in essence, perpendicular to each other in a geometric sense. To determine if two vectors are orthogonal, we look at their dot product. The dot product of two vectors is a way to multiply them together, and it gives a scalar (a single number) as a result. Mathematically, the dot product of vectors \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle \) is calculated as follows:

• \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3\)

These vectors are orthogonal if their dot product is zero. The reason for this is that when the dot product is zero, it means the angle between the vectors is 90 degrees. In three-dimensional space, this translates to the vectors being perpendicular. In our exercise, vectors (c) were determined to be orthogonal as their dot product was calculated to be zero. This means they do not influence each other's direction.

Parallel Vectors

When two vectors are parallel, they essentially point in the same or exact opposite directions. To check if vectors are parallel, we establish whether one vector is a scalar multiple of the other.

• For vectors \(\mathbf{u}\) and \(\mathbf{v}\), they are parallel if there exists a scalar \(k\) such that \(\mathbf{v} = k\mathbf{u}\).

When vectors are parallel, their magnitudes may differ, but the direction remains aligned. In our original exercise, vectors (a) were shown to be parallel, as each corresponding component of \(\mathbf{v}\) was a constant multiple of the corresponding component of \(\mathbf{u}\). By confirming consistent ratios across the vector components, we can verify their parallelism. This result means that the vectors lie along the same line in space albeit possibly scaled, and they will never intersect unless they are the same line.

Dot Product

The dot product is a fundamental operation in vector analysis that helps us glean important geometric information about vectors. In simpler terms, it provides insight into the angle between two vectors. The result is a scalar that is influenced by both the magnitudes of the vectors and the angle between them.

• The formula for the dot product of vectors \(\mathbf{u} = \langle u_1, u_2, u_3 \rangle\) and \(\mathbf{v} = \langle v_1, v_2, v_3 \rangle\) is: \(\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 \).

The dot product is widely used to determine

• Orthogonality: If \(\mathbf{u} \cdot \mathbf{v} = 0\), vectors are orthogonal.
• Projection: It assists in projecting one vector onto another.

In the context of our given vectors, the dot product was used to decide orthogonality and parallelism. It acts as a decisive tool for understanding these relational properties in vector space.