91Ó°ÊÓ

The 3D Cartesian coordinate system is a standard framework used to describe the position and movement of points and vectors in three-dimensional space. It extends the 2D Cartesian system into three dimensions by adding the z-axis, perpendicular to both the x and y-axes. Each point in this space can be represented as a combination of three coordinates:

• The x-coordinate, which measures a point's position along the horizontal axis.
• The y-coordinate, which measures vertical positioning.
• The z-coordinate, indicating depth relative to the observer.

These axes intersect at a central point referred to as the origin, denoted as (0, 0, 0). Here, vectors can have nonzero values along any of the three dimensions, making visualization in 3D essential. For instance, the original exercise describes vectors like \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = \mathbf{j} + \mathbf{k} \), which we imagine by moving 1 unit in the specified directions on the 3D Cartesian plane. This interpretation helps in analyzing vector relationships and conditions such as linear dependence or independence effectively.

Vector addition is a fundamental operation where two or more vectors are combined to produce a resultant vector. This process involves adding the corresponding components of the vectors involved. For vectors in a 3D Cartesian coordinate system, the components in the x, y, and z directions are added separately.
For example, consider vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \). Their sum, \( \mathbf{a} + \mathbf{b} \), is computed as follows:

• Add the i-components: \( a_1 + b_1 \).
• Add the j-components: \( a_2 + b_2 \).
• Add the k-components: \( a_3 + b_3 \).

The resultant vector is \( (a_1 + b_1)\mathbf{i} + (a_2 + b_2)\mathbf{j} + (a_3 + b_3)\mathbf{k} \). In the exercise, when we compute \( \mathbf{w} \) using vectors \( \mathbf{u} \) and \( \mathbf{v} \), we utilize vector addition by scaling \( \mathbf{u} \) and \( \mathbf{v} \) as necessary, and then summing their parts to explore scenarios such as \( \mathbf{w} = \mathbf{0} \) or specific values.

Vector components are the projections of a vector along the axes of a coordinate system. In a 3D Cartesian coordinate system, each vector is broken down into components along the x, y, and z axes. The components describe the influence of a vector in each direction, which is crucial for analyzing and manipulating vectors.
For instance, the vector \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) comprises components of 1 in both x and y directions, and 0 in the z direction. Similarly, \( \mathbf{v} = \mathbf{j} + \mathbf{k} \) has 0 in x, 1 in y, and 1 in z.

• The x-component allows us to see the horizontal extent of the vector.
• The y-component reveals the vector's vertical orientation.
• The z-component shows the vector's depth or out-of-plane aspect.

Understanding these components allows us to rewrite and manipulate vectors based on different conditions. For example, solving for \( \mathbf{w} = \mathbf{0} \) requires setting each component—x, y, and z—of \( \mathbf{w} \) to zero, leading to equations that help determine values of \( a \) and \( b \) in the given exercise.

Linear independence is a concept in vector spaces where a set of vectors is considered linearly independent if no vector in the set can be written as a linear combination of the others. This property helps us understand whether vectors provide unique direction without redundant or overlapping effects.
For vectors \( \mathbf{u} = \mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = \mathbf{j} + \mathbf{k} \), proving that a linear combination results in \( \mathbf{w} = \mathbf{0} \) requires checking whether non-zero coefficients \( a \) and \( b \) exist. In this case, both \( a \) and \( b \) must be zero for the combination to yield zero, affirming linear independence.

• If a vector can be described using others in the set, they are linearly dependent.
• Being linearly independent means each vector contributes new dimensionality to the space.

In the exercise, to show \( \mathbf{w} = \mathbf{0} \), the linear combination \( a\mathbf{u} + b\mathbf{v} \) leads us to equations like \( ai + aj + bj + bk = 0 \), which confirms dependence only when both parameters are zero.