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In vector mathematics, the dot product is a fundamental operation that helps determine important relationships between vectors. To compute the dot product of two vectors, simply multiply their corresponding components and sum up these products. Consider two vectors \ \( \vec{A} = A_x \hat{\imath} + A_y \hat{\jmath} \ \) and \ \( \vec{B} = B_x \hat{\imath} + B_y \hat{\jmath} \ \). Their dot product is calculated as:

• \( \vec{A} \cdot \vec{B} = A_x \cdot B_x + A_y \cdot B_y \).

The result of this operation is a scalar quantity, not a vector.
The dot product quantifies how much one vector goes in the direction of another. If the result is zero, the vectors are perpendicular to each other. If positive, they point roughly in the same direction, and if negative, they point in opposite directions.
In our exercise, knowing the dot product of vectors \( \vec{C} \) and \( \vec{B} \) allows us to use it in forming an equation to eventually find the components of vector \( \vec{C} \).

Vectors are perpendicular when they meet at a right angle, and a simple way to determine this is by using the dot product. If the dot product of two vectors is zero, it evidences their perpendicularity. This is because the cosine of the angle between them, which influences the calculation of the dot product, is zero when the angle is \(90^\circ\).
For example, two vectors \( \vec{A} \) and \( \vec{C} \) being perpendicular means that:

• \( \vec{A} \cdot \vec{C} = 0 \)

This relationship simplifies the task of determining whether two directions in space are at a right angle to one another.
In the given problem, vector \( \vec{C} \) is perpendicular to vector \( \vec{A} \), allowing us to set up the equation \( 5.0C_x - 6.5C_y = 0 \), as specified by the calculation of their dot product.

In mathematics, a system of equations is a set of two or more equations that share the same variables. Solving a system means finding all values of the variables that satisfy each equation in the system. There are various methods to solve systems, such as substitution or elimination.
In our exercise, two equations arise from the conditions outlined:

• \(5.0C_x - 6.5C_y = 0\)
• \(-3.5C_x + 7.0C_y = 15.0\)

To solve this system, we can use substitution:

• First, express \(C_x\) in terms of \(C_y\) using the first equation.
• Then substitute this expression into the second equation.
• Solve for \(C_y\) and back-substitute to find \(C_x\).

Using this method, we find the solution \(C_x \approx 7.95\) and \(C_y \approx 6.12\). With these values, vector \( \vec{C} \) satisfies both its perpendicularity with \( \vec{A} \) and the dot product condition with \( \vec{B} \). This example shows the power of systems of equations in solving complex real-world problems.