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Vector addition is a fundamental operation in mathematics, particularly in the study of vectors. It involves the process of adding two or more vectors to produce a single resultant vector. To add vectors, you simply add their corresponding components.
For example, if you have two vectors \( \mathbf{a} = \langle 2, 3 \rangle \) and \( \mathbf{b} = \langle 5, 4 \rangle \), then their sum \( \mathbf{a} + \mathbf{b} \) is calculated as follows:

• Add the first components: \( 2 + 5 = 7 \)
• Add the second components: \( 3 + 4 = 7 \)

So, \( \mathbf{a} + \mathbf{b} = \langle 7, 7 \rangle \).
This operation is simple yet crucial as it helps in finding resultant vectors, which can be important in physics, engineering, and computer graphics. Vector addition is also commutative, meaning \( \mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a} \), making calculations easier.

Vector scaling is the multiplication of a vector by a scalar, which results in a new vector. The scalar, which is simply a regular number, multiplies each component of the vector independently. This operation changes the magnitude of the vector without affecting its direction, unless the scalar is negative which reverses the direction.
For instance, given vector \( \mathbf{c} = \langle 6, -1 \rangle \), scaling this vector by 3 involves:

• Multiply the first component by 3: \( 6 \times 3 = 18 \)
• Multiply the second component by 3: \( -1 \times 3 = -3 \)

So, \( 3 \mathbf{c} = \langle 18, -3 \rangle \).
Vector scaling provides a way to adjust the size or length of vectors, which could represent force, velocity, or other physical quantities in applied contexts. It’s essential in operations like vector addition during computations.

A linear combination involves combining vectors using various scalars. It is a technique used to create new vectors or to express given vectors in terms of others. This concept is a cornerstone in linear algebra.
The expression \( c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + ... + c_n \mathbf{v}_n \) represents a linear combination, where \( c_1, c_2, ..., c_n \) are scalars and \( \mathbf{v}_1, \mathbf{v}_2, ..., \mathbf{v}_n \) are vectors.
In the given exercise, the expression \( 2\mathbf{a} + 4\mathbf{b} \) is a linear combination of vectors \( \mathbf{a} \) and \( \mathbf{b} \). Here, 2 and 4 are the scalars.

• Scale vector \( \mathbf{a} \) by 2 to get \( \langle 4, 6 \rangle \)
• Scale vector \( \mathbf{b} \) by 4 to get \( \langle 20, 16 \rangle \)
• Add these scaled vectors to get \( \langle 24, 22 \rangle \)

Linear combinations are essential because they allow us to express complicated problems in terms of simpler, ordered solutions, like basis vectors in space.

Vectors are essential objects in mathematics used to represent quantities that have both magnitude and direction. They can be used in 2-dimensional or 3-dimensional space, expressed typically as \( \mathbf{v} = \langle v_1, v_2, \ldots, v_n \rangle \). Each component of a vector represents a particular direction in a space, giving the vector its full dimensionality.
Vectors find applications in various fields such as physics, engineering, computer graphics, and more. They help describe a variety of phenomena:

• In physics, vectors are used to convey displacement, velocity, and acceleration.
• In engineering, they help model forces and moments.
• In computer graphics, vectors assist in navigating 3D models and scenes.

Understanding vectors enhances comprehension of directional quantities and operations such as vector addition and scaling. Mathematics curricula often introduce vectors to build foundational understanding that aids in solving practical problems elegantly.