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Adding vectors is a fundamental operation in vector mathematics. It involves taking two or more vectors and combining them to form a new vector. To perform vector addition, align vectors such that their corresponding dimensions are added together.

For example, consider two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \). To find their sum \( \mathbf{a} + \mathbf{b} \), just add their components:

• \( a_1 \) aligns with \( b_1 \) for one dimension.
• \( a_2 \) aligns with \( b_2 \) for the other dimension.

The resulting vector is \( \langle a_1 + b_1, a_2 + b_2 \rangle \).

Returning to our example from the exercise, after scaling vectors \( \mathbf{u} \) and \( \mathbf{v} \), they were \( \langle \frac{9}{5}, -\frac{6}{5} \rangle \) and \( \langle -\frac{8}{5}, 4 \rangle \), respectively. Their vector addition results in \( \langle \frac{1}{5}, \frac{14}{5} \rangle \), obtained by component-wise addition.

Scalar multiplication involves multiplying a vector by a scalar (a real number) which adjusts the vector's magnitude without changing its direction. It modifies every component of the vector proportionally.

For a vector \( \mathbf{v} = \langle x, y \rangle \) and a scalar \( c \), the scalar multiplication is calculated as follows:

• Multiply each component of the vector by the scalar: \( c \cdot \mathbf{v} = \langle c \cdot x, c \cdot y \rangle \)

The operation effectively stretches or compresses the vector.

In the provided exercise, \( \frac{3}{5} \) was multiplied by \( \mathbf{u} = \langle 3, -2 \rangle \), resulting in \( \langle \frac{9}{5}, -\frac{6}{5} \rangle \), and \( \frac{4}{5} \) was multiplied by \( \mathbf{v} = \langle -2, 5 \rangle \), which gives \( \langle -\frac{8}{5}, 4 \rangle \). These scaled vectors are then used in vector addition.

The magnitude of a vector, or its length, represents how long the vector is in the vector space. It is akin to finding the length of the hypotenuse in a right triangle, using the Pythagorean theorem.

For a two-dimensional vector \( \mathbf{a} = \langle x, y \rangle \), the magnitude is given by calculating the square root of the sum of the squares of its components:

• \( |\mathbf{a}| = \sqrt{x^2 + y^2} \)

This formula provides a straightforward way to compute the vector's length.

In the exercise, the magnitude was calculated for the vector \( \langle \frac{1}{5}, \frac{14}{5} \rangle \) resulting from previous operations. Applying the formula gives \( |\mathbf{w}| = \sqrt{\left(\frac{1}{5}\right)^2 + \left(\frac{14}{5}\right)^2} = \frac{\sqrt{197}}{5} \). This demonstrates how to determine the vector's overall magnitude.