Let A=( a1 a2 a3 b1 b2 b3 c1 c2 c3 ) and v̅1=a1 i̅+b1 j̅+c1 k̅

step 1 So in the given question we have two statements that are given in the question and the first sta

Let A=( a1 a2 a3 b1 b2 b3 c1 ...

$$ \text { Let } A=\left(\begin{array}{lll} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array}\right) $$ and $\bar{v}_{1}=a_{1} \bar{i}+b_{1} \bar{j}+c_{1} \bar{k}$ $$ \begin{array}{l} \overline{\mathrm{v}_{2}}=\mathrm{a}_{2} \overline{\mathrm{i}}+\mathrm{b}_{2} \overline{\mathrm{j}}+\mathrm{c}_{2} \overline{\mathrm{k}} \\ \overline{\mathrm{v}_{3}}=\mathrm{a}_{3} \overline{\mathrm{i}}+\mathrm{b}_{3} \overline{\mathrm{j}}+\mathrm{c}_{3} \overline{\mathrm{k}} \end{array} $$ be mutually orthogonal vectors.

$$
\text { Let } A=\left(\begin{array}{lll}
a_{1} & a_{2} & a_{3} \\
b_{1} & b_{2} & b_{3} \\
c_{1} & c_{2} & c_{3}
\end{array}\right)
$$
and $\bar{v}_{1}=a_{1} \bar{i}+b_{1} \bar{j}+c_{1} \bar{k}$
$$
\begin{array}{l}
\overline{\mathrm{v}_{2}}=\mathrm{a}_{2} \overline{\mathrm{i}}+\mathrm{b}_{2} \overline{\mathrm{j}}+\mathrm{c}_{2} \overline{\mathrm{k}} \\
\overline{\mathrm{v}_{3}}=\mathrm{a}_{3} \overline{\mathrm{i}}+\mathrm{b}_{3} \overline{\mathrm{j}}+\mathrm{c}_{3} \overline{\mathrm{k}}
\end{array}
$$
be mutually orthogonal vectors.

Key Concepts

Matrix Representation of Vectors

A matrix can be used to organize several vectors, either as rows or columns. When the columns of a matrix represent vectors, the matrix encapsulates these components in a structured format that facilitates operations like testing for orthogonality among the vectors. This is often applied in problems related to projections, rotations, and the study of vector spaces.

Vector Representation in Coordinate Systems

Representing vectors in a coordinate system involves expressing them as linear combinations of standard unit vectors (for example, i, j, k in ?³). This representation simplifies many vector operations, such as calculating the dot product or cross product, and is the basis for understanding more complex concepts like linear transformations and changes of basis.

Orthogonality

Orthogonality refers to the relationship between vectors that are perpendicular to each other. In an inner product space, such as ?³, two vectors are orthogonal if their dot product equals zero. This concept is central in linear algebra and plays a key role in areas such as projections, decompositions, and simplifying computations in various multidimensional problems.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. It is calculated as the sum of the products of corresponding components of two vectors. The dot product is particularly important because it provides a concise test for orthogonality; if the dot product of two vectors is zero, the vectors are orthogonal.

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