91Ó°ÊÓ

When we talk about vector addition, we're referring to the process of combining two vectors to form a new vector. This is done by adding each corresponding component from the vectors involved. For instance, given vectors \( \mathbf{a} = \langle 2, 3, 3 \rangle \) and \( \mathbf{b} = \langle 0, 1, 4 \rangle \), vector addition can be visualized as follows:

• The first component of the resulting vector is \(2 + 0 = 2\).
• The second component is \(3 + 1 = 4\).
• The third component is \(3 + 4 = 7\).

Hence, the sum of these vectors is \( \mathbf{a} + \mathbf{b} = \langle 2, 4, 7 \rangle \). The process is straightforward: just pair up each entry, add them, and you're good to go. Vector addition is commutative, meaning the order of addition does not affect the result: \( \mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a} \). Understanding vector addition is important because it shows how two force vectors, for example, can combine to create an equivalent single force in physics.

Vector subtraction operates similarly to vector addition but involves subtracting the corresponding components of one vector from another. It's best understood by considering it as the addition of a negative vector. Given vectors \( \mathbf{a} = \langle 2, 3, 3 \rangle \) and \( \mathbf{b} = \langle 0, 1, 4 \rangle \), subtracting \( \mathbf{b} \) from \( \mathbf{a} \) can be demonstrated as:

• First component: \(2 - 0 = 2\)
• Second component: \(3 - 1 = 2\)
• Third component: \(3 - 4 = -1\)

Thus, the result of the subtraction is \( \mathbf{a} - \mathbf{b} = \langle 2, 2, -1 \rangle \). Essentially, vector subtraction tells you the vector that would "move" from \( \mathbf{b} \) to \( \mathbf{a} \). In physics, it's akin to finding the relative displacement when two vectors denote different positions or states.

Scalar multiplication is a key concept in vectors that involves multiplying every component of a vector by a scalar (a real number). This operation either scales (enlarges or shrinks) the vector or changes its direction if the scalar is negative. For example, when given a vector \(\mathbf{a} = \langle 2, 3, 3 \rangle\) and a scalar 2, scalar multiplication proceeds as follows:

• Multiply the first component: \( 2 \times 2 = 4 \)
• Multiply the second component: \( 2 \times 3 = 6 \)
• Multiply the third component: \( 2 \times 3 = 6 \)

As a result, \( 2\mathbf{a} = \langle 4, 6, 6 \rangle \). Scalar multiplication stretches the vector if the scalar is greater than 1 or compresses it if the scalar is between 0 and 1. If the scalar is negative, the vector is flipped in direction. This operation is essential in defining the magnitude of vectors and plays a significant role in vector fields and linear transformations.

Vectors in three dimensions extend the concept of vectors on a plane to the 3D space, commonly denoted as \((x, y, z)\). These vectors are essential in fields like physics and engineering because they can represent points or directions in 3D space, such as velocity or forces acting on a body. In our exercise, we dealt with 3D vectors \(\mathbf{a} = \langle 2, 3, 3 \rangle\) and \(\mathbf{b} = \langle 0, 1, 4 \rangle\), which show examples of vectors that include components along three axes: x, y, and z. When performing operations like addition, subtraction, or scalar multiplication on 3D vectors, the concepts are similar to those in two dimensions, just with the added z-component. All operations we demonstrated, such as \( \mathbf{a} + \mathbf{b} \) or \( 2\mathbf{a} \), take place independently within each coordinate, maintaining the vector's integrity as a unit. Understanding vectors in three dimensions is vital because they help model and solve real-world problems that occur in spatial contexts. Whether calculating trajectories, forces, or fields, 3D vectors are foundational in capturing the full picture.