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Scalar multiplication is a fundamental operation in vector algebra. It involves multiplying a vector by a scalar (a real number), resulting in a new vector whose magnitude is scaled by the scalar, while the direction remains the same unless the scalar is negative.

For the vector \( \mathbf{u} = \langle u_1, u_2 \rangle \), multiplying it by a scalar \( c \) gives:\[ c \mathbf{u} = \langle c \cdot u_1, c \cdot u_2 \rangle\]In our example, calculating \( -\frac{5}{13}\mathbf{u} \) involves multiplying each component of \( \mathbf{u} = \langle 3, -2 \rangle \) by \(-\frac{5}{13}\):

• \( -\frac{5}{13} \times 3 = -\frac{15}{13} \)
• \( -\frac{5}{13} \times (-2) = \frac{10}{13} \)

Similarly, multiplying vector \( \mathbf{v} = \langle -2, 5 \rangle \) by \( \frac{12}{13} \):

• \( \frac{12}{13} \times (-2) = -\frac{24}{13} \)
• \( \frac{12}{13} \times 5 = \frac{60}{13} \)

Scalar multiplication is quite intuitive once understood and plays a significant role in scaling vectors in various mathematical and physical contexts.

The magnitude of a vector, often referred to as its length, measures how long the vector is. It's always a non-negative number and can be calculated using the Euclidean distance formula.

For a vector \( \mathbf{a} = \langle x, y \rangle \), the magnitude \( \|\mathbf{a}\| \) is computed as:\[\|\mathbf{a}\| = \sqrt{x^2 + y^2}\]This effectively applies the Pythagorean theorem to vectors, allowing us to find their length.

In our problem, after computing the resultant vector \( \langle -3, \frac{70}{13} \rangle \), the magnitude is:

• Calculate \((-3)^2 = 9 \)
• Calculate \(\left(\frac{70}{13}\right)^2 = \frac{4900}{169} \)
• Add these results: \( 9 + \frac{4900}{169}\)
• This simplifies to \( \frac{6421}{169} \)
• And the final magnitude is \( \frac{\sqrt{6421}}{13} \)

The calculation of a vector's magnitude is essential in various fields such as physics, engineering, and computer graphics, where the distance or size of a vector is required.

Vectors can be expressed in different forms, with the component form being one of the most common. It lists the vector's coordinates or components in a simple bracket notation, making it easy to use in calculations and visualizations.

The component form of any vector \( \mathbf{a} \) in two-dimensional space is given as \( \langle a_1, a_2 \rangle \), where \( a_1 \) and \( a_2 \) represent the horizontal and vertical components, respectively.

To find the component form of a sum of vectors, such as \(-\frac{5}{13} \mathbf{u} + \frac{12}{13} \mathbf{v} \), you would separately compute the scalar multiplication of each vector and then add the results:

Results of scalar multiplication:

• \( \left(-\frac{15}{13}, \frac{10}{13}\right) \) from \( -\frac{5}{13} \mathbf{u} \)
• \( \left(-\frac{24}{13}, \frac{60}{13}\right) \) from \( \frac{12}{13} \mathbf{v} \)

Add these component-wise:

• Horizontal: \( -\frac{15}{13} + \left(-\frac{24}{13}\right) = -\frac{39}{13} = -3 \)
• Vertical: \( \frac{10}{13} + \frac{60}{13} = \frac{70}{13} \)

So, the component form of the vector is \( (-3, \frac{70}{13}) \). Understanding this form makes it simpler to perform other vector operations like dot product or cross product in higher dimensions.