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Vectors are mathematical objects used to represent quantities that have both a magnitude and a direction. In the realm of physics and engineering, they are crucial. Notably, vectors are often depicted in terms of their components along the standard basis vectors, usually denoted as \( \vec{i} \) and \( \vec{j} \). These basis vectors are unit vectors: \( \vec{i} \) points in the horizontal direction, and \( \vec{j} \) in the vertical direction.

For instance, a vector \( 2 \vec{i} + 3 \vec{j} \) indicates that the vector has a component of 2 in the direction of \( \vec{i} \) and a component of 3 in the direction of \( \vec{j} \). This is an important way to break down vectors, allowing for easier manipulation and understanding in problems like the one given in the exercise. By expressing vectors in terms of these components, operations such as addition and subtraction become more straightforward.

The distributive property is a fundamental property of arithmetic and algebra that allows one to distribute a factor across terms inside a parenthesis. This was a crucial step in solving the given vector problem.

In this context, applying distributive property means we will multiply \(-3\) by each component of the vector \( (2 \vec{i} + \vec{j}) \). Here is how it simplifies:

• \((-3)(2 \vec{i}) = -6 \vec{i}\)
• \((-3)(\vec{j}) = -3 \vec{j}\)

This allows us to rewrite the problem in a more manageable form: \((\vec{i} + 2 \vec{j}) - 6 \vec{i} - 3 \vec{j} \). It demonstrates how values can be expanded and simplified according to this reliable property. Mastering the distributive property is essential for dealing with vector expressions and more complex algebraic scenarios.

Combining like terms is a method that simplifies expressions by merging terms with the same variables. In vector arithmetic, you apply it to each vector component separately.

For vectors, this process is akin to adding together portions going in the same direction. In our example, you have:

• \(\vec{i} - 6\vec{i}\) which equals \(-5\vec{i}\).
• \(2\vec{j} - 3\vec{j}\) which equals \(-1\vec{j}\).

This results in the final simplified form, \(-5\vec{i} - \vec{j}\).

Remember, when combining, focus on each term In isolation. It reduces complexity and chances for mistakes. Approach vector addition as a method to find a single, resultant vector by summing each corresponding component directionally.