Determine whether the given vectors are orthogonal,...

Determine whether the given vectors are orthogonal, parallel, or neither. $$ \begin{array}{l}{\text { (a) } \mathbf{u}=\langle- 5,4,-2\rangle, \quad \mathbf{v}=\langle 3,4,-1\rangle} \\ {\text { (b) } \mathbf{u}= 9 \mathbf{i}-6 \mathbf{j}+3 \mathbf{k}, \quad \mathbf{v}=-6 \mathbf{i}+4 \mathbf{j}-2 \mathbf{k}} \\ {\text { (c) } \mathbf{u}=\langle c, c, c\rangle, \quad \mathbf{v}=\langle c, 0,-c\rangle}\end{array} $$

Determine whether the given vectors are orthogonal, parallel, or neither.
$$
\begin{array}{l}{\text { (a) } \mathbf{u}=\langle- 5,4,-2\rangle, \quad \mathbf{v}=\langle 3,4,-1\rangle} \\ {\text { (b) } \mathbf{u}= 9 \mathbf{i}-6 \mathbf{j}+3 \mathbf{k}, \quad \mathbf{v}=-6 \mathbf{i}+4 \mathbf{j}-2 \mathbf{k}} \\ {\text { (c) } \mathbf{u}=\langle c, c, c\rangle, \quad \mathbf{v}=\langle c, 0,-c\rangle}\end{array}
$$

Key Concepts

Dot Product

The dot product of two vectors is a fundamental operation in vector algebra obtained by multiplying corresponding components and summing the results. It provides a measure of how much two vectors are aligned with each other, and it is central to determining whether vectors are orthogonal or share other relationships.

Orthogonality

Vectors are considered orthogonal if their dot product is zero. This condition signifies that the vectors meet at a right angle, which is a key concept in understanding perpendicularity in vector spaces.

Parallelism

Vectors are parallel if one vector is a scalar multiple of the other. This means that they have the same or exact opposite directions. Checking for parallelism involves verifying whether each component of one vector is proportional to the corresponding component of the other vector.

Scalar Multiplication and Proportionality

Determining whether one vector is a scalar multiple of another is an important technique to check for parallelism. If a constant exists such that multiplying one vector by that constant yields the other vector, the vectors are parallel. Otherwise, if the dot product is not zero and the vectors are not scalar multiples, they are neither orthogonal nor parallel.

Parametric Representation of Vectors

When vectors are expressed in terms of a parameter, like a common scalar in each component, analyzing their relationships requires careful handling of that parameter. Understanding how the parameter affects the vector directions is critical for determining conditions such as orthogonality or parallelism in an abstract sense.

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