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The dot product is a fundamental operation in vector mathematics. It provides a way to multiply two vectors to yield a scalar value. This scalar indicates how much one vector acts in the direction of another.
The dot product of two vectors \(\vec{a} = (a_1, a_2, a_3)\) and \(\vec{b} = (b_1, b_2, b_3)\) is calculated as:

• \(a_1 \cdot b_1\)
• \(a_2 \cdot b_2\)
• \(a_3 \cdot b_3\)

All these terms are summed up to get the dot product: \(a_1 \cdot b_1 + a_2 \cdot b_2 + a_3 \cdot b_3\).
This product becomes zero when the vectors are perpendicular to each other. Understanding the dot product is essential for solving problems involving perpendicular vectors.

The algebraic condition for two vectors to be perpendicular is that their dot product must equal zero. Once you set up the dot product equation, such as with vectors \(\vec{c}=(-3, p, -1)\) and \(\vec{d}=(1, -4, q)\), the condition requires rearranging the dot product calculation to zero:
\[-3 \cdot 1 + p \cdot (-4) + (-1) \cdot q = 0\]
This results in the equation:

• \(-3 - 4p - q = 0\)

To satisfy the perpendicularity condition and solve for specific vector components, you manipulate this equation. Perpendicular vectors manipulate the space they occupy, and their algebraic conditions help define this essential characteristic.

Vector components refer to the individual values that make up a vector. In a three-dimensional space, a vector \(\vec{a}\) is expressed in terms of its components: \(\vec{a} = (a_1, a_2, a_3)\). Each component indicates how far the vector stretches along respective axes.
In our specific exercise, the components \((-3, p, -1)\) and \((1, -4, q)\) are important. They influence the behavior of the vector, particularly their direction and magnitude. To maintain the perpendicularity condition, adjusting components such as \(p\) or \(q\) is often necessary.
By substituting specific values, like setting \(q = -3\), you can solve for unknown components, such as \(p\), keeping the vectors perpendicular. This manipulation of vector components allows for precise control over the vectors' relationships with each other.