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Correct Answer: a1=b1,a2=b2,a3=b3
in the cartesian coordinate system, vectors can be described in terms of their components along the basis vectors. the basis vectors for a three-dimensional cartesian system are typically denoted as \( \mathbf{e}_1 = (1,0,0) \), \( \mathbf{e}_2 = (0,1,0) \), and \( \mathbf{e}_3 = (0,0,1) \). each vector in this space can be represented as a linear combination of these basis vectors.
given two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), expressed in terms of the basis vectors:
- \( \mathbf{a} = a_1\mathbf{e}_1 + a_2\mathbf{e}_2 + a_3\mathbf{e}_3 \)
- \( \mathbf{b} = b_1\mathbf{e}_1 + b_2\mathbf{e}_2 + b_3\mathbf{e}_3 \)
where \( a_1, a_2, a_3 \) and \( b_1, b_2, b_3 \) are the components of vectors \( \mathbf{a} \) and \( \mathbf{b} \) respectively.
to determine if vectors \( \mathbf{a} \) and \( \mathbf{b} \) are equal, we need to check equality of their respective components along each of the basis vectors. this is because vectors in a vector space are equal if and only if their corresponding components along each basis direction are equal. thus, for vectors \( \mathbf{a} \) and \( \mathbf{b} \) to be equal:
- \( a_1 = b_1 \)
- \( a_2 = b_2 \)
- \( a_3 = b_3 \)
these component-wise equalities must all be true simultaneously for \( \mathbf{a} \) to be considered equal to \( \mathbf{b} \). in the symbolic form, the equation to show that vectors \( \mathbf{a} \) and \( \mathbf{b} \) are equal can be written as:
\[ \mathbf{a} = \mathbf{b} \iff (a_1 = b_1) \land (a_2 = b_2) \land (a_3 = b_3) \]
it is important to note that the equation \( a_1b_1 + a_2b_2 + a_3b_3 = 0 \) mentioned in the question does not imply that the vectors \( \mathbf{a} \) and \( \mathbf{b} \) are equal. instead, this equation usually describes a condition where vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal (perpendicular to each other), assuming that neither vector is the zero vector. hence, this equation is not relevant for checking the equality of \( \mathbf{a} \) and \( \mathbf{b} \) but rather their orthogonality.
in summary, for vectors \( \mathbf{a} \) and \( \mathbf{b} \) in a cartesian coordinate system to be equal, their corresponding scalar components along each of the basis vectors must be identical. this is a direct application of the definition of vector equality in linear algebra.
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