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Vector addition is a fundamental operation where two vectors are combined to produce another vector. To add two vectors, you add their corresponding components. If you have vectors \(\textbf{A} = \langle a_1, a_2, a_3 \rangle\) and \(\textbf{B} = \langle b_1, b_2, b_3 \rangle\), their sum is: \(\textbf{A} + \textbf{B} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3 \rangle\).
For example, if \(\textbf{A} = \langle 1, 2, 3 \rangle\) and \(\textbf{B} = \langle 4, -3, -1 \rangle\), then their sum would be: \(\textbf{A} + \textbf{B} = \langle 1 + 4, 2 + (-3), 3 + (-1) \rangle = \langle 5, -1, 2 \rangle\).
This operation is simple but crucial as it is the basis for more complex vector operations.

Scalar multiplication involves multiplying a vector by a scalar (a single number). Each component of the vector is multiplied by this scalar. If you have a vector \(\textbf{A} = \langle a_1, a_2, a_3 \rangle\) and a scalar \( k \), the result is another vector: \(\textbf{A}_{mult} = k \cdot \textbf{A} = \langle k \cdot a_1, k \cdot a_2, k \cdot a_3 \rangle\).
Take vector \(\textbf{A} = \langle 1, 2, 3 \rangle\) and scalar 2, you get: \(\textbf{A}_{mult} = 2 \cdot \textbf{A} = \langle 2 \cdot 1, 2 \cdot 2, 2 \cdot 3 \rangle = \langle 2, 4, 6 \rangle\).
Scalar multiplication is useful in scaling vectors, making them longer or shorter while preserving their direction.

Vector subtraction is another essential operation closely related to vector addition. It involves subtracting the components of one vector from another. Given vectors \(\textbf{A} = \langle a_1, a_2, a_3 \rangle\) and \(\textbf{B} = \langle b_1, b_2, b_3 \rangle\), the result is: \(\textbf{A} - \textbf{B} = \langle a_1 - b_1, a_2 - b_2, a_3 - b_3 \rangle\).
For instance, if \(\textbf{A} = \langle 2, 4, 6 \rangle\) and \(\textbf{C} = \langle -5, -3, 5 \rangle\), then: \(\textbf{A} - \textbf{C} = \langle 2 - (-5), 4 - (-3), 6 - 5 \rangle = \langle 2 + 5, 4 + 3, 6 - 5 \rangle = \langle 7, 7, 1 \rangle\).
Understanding vector subtraction is crucial for finding differences between vectors and solving related problems.