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The horizontal component of a vector represents its projection along the x-axis. To calculate this component, we utilize the cosine function, which helps determine how much of the vector's direction falls along the horizontal line.
In our scenario, given a vector with a length \( |\mathbf{v}| = \sqrt{3} \) and a direction angle of \( \theta = 300^{\circ} \), the horizontal component can be expressed as follows:

• Utilize the formula \( x = |\mathbf{v}| \cdot \cos(\theta) \).
• Substitute the given values into the formula to find \( x = \sqrt{3} \cdot \cos(300^{\circ}) \).
• Since \( \cos(300^{\circ}) = \frac{1}{2} \), compute \( x = \sqrt{3} \cdot \frac{1}{2} = \frac{\sqrt{3}}{2} \).

The result, \( x = \frac{\sqrt{3}}{2} \), tells us the magnitude of this vector's horizontal influence.

In contrast to the horizontal component, the vertical component of a vector shows how much of the vector lies along the y-axis. Calculating this requires the sine function, which measures the projection of the vector in the vertical direction.
For our given vector of length \( |\mathbf{v}| = \sqrt{3} \) and direction angle \( \theta = 300^{\circ} \):

• Apply the formula \( y = |\mathbf{v}| \cdot \sin(\theta) \).
• Insert the values into this formula to find \( y = \sqrt{3} \cdot \sin(300^{\circ}) \).
• Recognize that \( \sin(300^{\circ}) = -\frac{\sqrt{3}}{2} \), leading to the calculation \( y = \sqrt{3} \cdot \left(-\frac{\sqrt{3}}{2}\right) = -\frac{3}{2} \).

Thus, the component \( y = -\frac{3}{2} \) reveals the vertical influence of the vector, directing downwards due to its negative sign.

Unit vectors are a fundamental concept in vector mathematics. They are vectors with a magnitude of 1, used primarily to indicate direction along axes without influencing the vector's length. In two-dimensional space, the standard unit vectors are \( \mathbf{i} \) for the horizontal x-axis and \( \mathbf{j} \) for the vertical y-axis.
A vector expressed in terms of unit vectors provides a clear breakdown of its components. For our vector with horizontal \( x = \frac{\sqrt{3}}{2} \) and vertical \( y = -\frac{3}{2} \) components, it can be written using unit vectors as follows:

• The vector \( \mathbf{v} = x \mathbf{i} + y \mathbf{j} \).
• Substitute the known components into the equation to yield \( \mathbf{v} = \frac{\sqrt{3}}{2} \mathbf{i} - \frac{3}{2} \mathbf{j} \).

This representation makes it evident how much of the vector's impact is in each primary direction, harnessing unit vectors as a simple yet effective tool for depicting a vector's attributes.