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Unit vectors are vectors with a magnitude of one. They are often used to denote direction. In two-dimensional space, the most common unit vectors are \( \mathbf{i} \) and \( \mathbf{j} \). These represent the standard basis vectors for the x and y axes respectively:

• \( \mathbf{i} \) has a direction along the x-axis and can be expressed as \((1, 0)\).
• \( \mathbf{j} \) has a direction along the y-axis and can be expressed as \((0, 1)\).

Unit vectors are crucial in simplifying vector expressions and performing operations like vector addition and scalar multiplication. When we express a vector in terms of unit vectors, we clearly identify how much of each direction (x and y in this case) contributes to the vector's overall magnitude and direction. For example, in the vector \(19\mathbf{i} + 23\mathbf{j}\), 19 represents how many units of \( \mathbf{i} \) the vector contains, and 23 represents how many units of \( \mathbf{j} \) it contains.

Scalar multiplication involves multiplying a vector by a scalar, which is a real number. This operation changes the magnitude of the vector but not its direction. To perform scalar multiplication:

• Multiply each component of the vector by the scalar.
• The resulting vector has components that are scaled versions of the original.

In our exercise, we see scalar multiplication in action:

• For the vector \((5, 3)\), multiplying by the scalar 3 results in \((15, 9)\).
• For the vector \((2, 7)\), multiplying by the scalar 2 results in \((4, 14)\).

Thus, scalar multiplication can be viewed as stretching or compressing the vector based on the scalar's value. It maintains the vector's direction unless the scalar is negative, which would invert its direction.

Vector notation is a way of representing vectors using either component form or unit vector form.

• Component Form: Vectors are expressed using ordered pairs or triples, depending on the dimension, such as \((x, y)\) in 2D space or \((x, y, z)\) in 3D space.
• Unit Vector Form: Vectors are expressed in terms of the standard unit vectors, which improves clarity especially in physics and engineering. For example, \( \mathbf{i} \) represents the unit vector along the x-axis, \( \mathbf{j} \) represents the unit vector along the y-axis, and \( \mathbf{k} \) is used for the z-axis in three dimensions.

Using unit vector form has the advantage of clearly showing the contribution of each component direction to the vector. In the exercise, we converted the component form \((19, 23)\) to unit vector notation \(19\mathbf{i} + 23\mathbf{j}\) to express the vector in a more descriptive manner.This approach aids in visualization and understanding the impact of each component in the computed vector.