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Unit vector notation is a way to express vectors in terms of their components along the coordinate axes, usually using symbols like \( \hat{\mathrm{i}} \), \( \hat{\mathrm{j}} \), and \( \hat{\mathrm{k}} \). Each symbol represents a unit vector, which has a magnitude of 1 and points in the direction of one of the coordinate axes. For example, \( \hat{\mathrm{i}} \) represents a unit vector along the x-axis, while \( \hat{\mathrm{j}} \) is along the y-axis.

In the given exercise, vector \( \vec{d}_3 \) is expressed in unit vector notation as \( 2 \hat{\mathrm{i}} + 4 \hat{\mathrm{j}} \). This means \( \vec{d}_3 \) has a component of 2 units in the x-direction and 4 units in the y-direction. By using unit vector notation, it's easy to identify and work with vector components. When solving vector problems, having vectors expressed in this form allows us to directly calculate their sum or difference by combining like components.

Vector addition involves combining two or more vectors to produce a resultant vector. When using unit vector notation, this is done by simply adding together the corresponding components of the vectors involved. Let's say we have two vectors, \( \vec{a} = a_x \hat{\mathrm{i}} + a_y \hat{\mathrm{j}} \) and \( \vec{b} = b_x \hat{\mathrm{i}} + b_y \hat{\mathrm{j}} \). Their sum, \( \vec{c} = \vec{a} + \vec{b} \), will be
\[\vec{c} = (a_x + b_x) \hat{\mathrm{i}} + (a_y + b_y) \hat{\mathrm{j}}\]

In the original problem, we needed to find \( \vec{d}_1 + \vec{d}_2 \). After substituting the values and understanding the components of \( \vec{d}_3 \), and subsequently \( \vec{d}_1 \) and \( \vec{d}_2 \), we find that by adding the vectors, we handle the i and j components separately. This way, we can efficiently solve the equation \( \vec{d}_1 + \vec{d}_2 = 10 \hat{\mathrm{i}} + 20 \hat{\mathrm{j}} \).

Vector subtraction is similar to vector addition, but instead, it involves finding the difference between two vectors. This operation subtracts the corresponding components of one vector from another. Using the same vector representation as in addition, for vectors \( \vec{a} = a_x \hat{\mathrm{i}} + a_y \hat{\mathrm{j}} \) and \( \vec{b} = b_x \hat{\mathrm{i}} + b_y \hat{\mathrm{j}} \), the difference \( \vec{d} = \vec{a} - \vec{b} \) is
\[\vec{d} = (a_x - b_x) \hat{\mathrm{i}} + (a_y - b_y) \hat{\mathrm{j}}\]

In the exercise, vector subtraction provided valuable insights with \( \vec{d}_1 - \vec{d}_2 \) being set to \( 6 \hat{\mathrm{i}} + 12 \hat{\mathrm{j}} \). By subtracting, we were able to decouple the problem into simpler components. This methodical approach allows us to solve the vector equations, isolating each vector and finding their direct values in unit vector notation.