91Ó°ÊÓ

Vectors are essential components in mathematics, particularly in fields like physics and engineering. A vector is a quantity that has both magnitude and direction, unlike a scalar which only possesses magnitude.
Vectors are typically represented in a coordinate system, for example, as \( \langle a, b \rangle \) in two dimensions or \( \langle a_1, a_2, a_3 \rangle \) in three dimensions.
These coordinates define the vector's direction and magnitude from the origin in space.

• The notation \( \langle a, b \rangle \) denotes a vector with components \( a \) and \( b \) along the x and y axes, respectively.
• The length or size of a vector is known as its magnitude, calculated using the Pythagorean theorem for two-dimensional vectors.

Having a clear understanding of vectors is crucial as they are the building blocks for more complex operations such as vector addition, subtraction, and scalar multiplication.

Vectors have several distinctive properties that facilitate numerous mathematical operations. These properties provide a foundation for comprehension and application in various scenarios.
One key property is the **additive property**, which states that the sum of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) results in a new vector: \( \mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle \).
Another significant attribute is the **commutative property**: the order of vector addition does not change the result, meaning \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \). Other important vector properties include:

• **Associative property**: The way vectors are grouped in addition does not affect the result \( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \).
• **Zero vector property**: Adding the zero vector \( \mathbf{0} \) to a vector \( \mathbf{v} \) does not change the vector: \( \mathbf{v} + \mathbf{0} = \mathbf{v} \).
• **Negative vector property**: Each vector has an additive inverse, allowing \( \mathbf{v} + (-\mathbf{v}) = \mathbf{0} \).

Understanding these properties allows for effective manipulation and transformation of vectors in mathematics.

Scalar multiplication involves scaling a vector by a real number known as a scalar. This process alters the vector’s magnitude but maintains its direction unless the scalar is negative, which reverses the direction.
For a vector \( \mathbf{v} = \langle a, b \rangle \) and a scalar \( k \), the resulting vector from multiplying the vector by the scalar is \( k\mathbf{v} = \langle ka, kb \rangle \).
This operation results in a new vector whose magnitude is \( |k| \) times the original magnitude. Importantly, scalar multiplication retains the vector's original direction (or reverses it if negative).

• If \( k = 1 \), the vector remains unchanged: \( 1\mathbf{v} = \mathbf{v} \).
• If \( k = 0 \), the result is the zero vector: \( 0\mathbf{v} = \mathbf{0} \).
• If \( |k| > 1 \), the vector lengthens, and if \( |k| < 1 \), it shortens.

Scalar multiplication is a foundational operation necessary for understanding more advanced mathematical and physical concepts.