Angle Between Vector Q And Resultant Of (2Q + 2P) And (2Q - 2P)

Find the angle between vector ${ \vec{Q} }$ and the resultant of ${ (2\vec{Q} + 2\vec{P}) }$ and ${ (2\vec{Q} - 2\vec{P}) }$.

In the realm of vector mathematics, a fundamental concept involves determining the angle between vectors and their resultants. This exploration delves into a specific problem that epitomizes this concept: finding the angle between a vector ${ \vec{Q} }$ and the resultant of two vector expressions, ${ (2\vec{Q} + 2\vec{P}) }$ and ${ (2\vec{Q} - 2\vec{P}) }$. This problem not only reinforces our understanding of vector addition but also highlights the significance of trigonometric functions in determining angles in vector spaces. We will systematically dissect the problem, employing vector algebra principles and trigonometric relationships to arrive at the correct solution. This article serves as a comprehensive guide to understanding this specific vector problem and provides a broader understanding of vector mathematics.

The problem at hand asks us to find the angle between the vector ${ \vec{Q} }$ and the resultant of the vectors ${ (2\vec{Q} + 2\vec{P}) }$ and ${ (2\vec{Q} - 2\vec{P}) }$. To solve this, we need to first determine the resultant vector by adding the two given vectors. Once we have the resultant vector, we can use the dot product formula to find the angle between ${ \vec{Q} }$ and the resultant. The dot product provides a way to relate the magnitudes of the vectors and the cosine of the angle between them. By applying the dot product and then using the inverse trigonometric function, specifically the arctangent, we can precisely calculate the angle. This problem encapsulates the core principles of vector algebra and trigonometry, providing an excellent opportunity to reinforce these concepts. Understanding the relationship between vectors and their resultants is crucial in various fields, including physics and engineering, where vector analysis is extensively used. This problem's methodology can be applied to more complex vector problems, making it a valuable exercise in mastering vector operations.

To embark on this problem, our initial step involves determining the resultant vector. This is achieved by performing vector addition of the given vectors, ${ (2\vec{Q} + 2\vec{P}) }$ and ${ (2\vec{Q} - 2\vec{P}) }$. Vector addition is a fundamental operation in vector algebra, where corresponding components of the vectors are added together. Mathematically, the resultant vector, denoted as ${ \vec{R} }$, is expressed as: ${ \vec{R} = (2\vec{Q} + 2\vec{P}) + (2\vec{Q} - 2\vec{P}) }$ Simplifying this expression, we combine like terms, specifically the terms involving ${ \vec{Q} }$ and ${ \vec{P} }$. This process involves adding the coefficients of the respective vectors. The addition of the ${ \vec{Q} }$ terms yields ${ 2\vec{Q} + 2\vec{Q} = 4\vec{Q} }$, while the addition of the ${ \vec{P} }$ terms results in ${ 2\vec{P} - 2\vec{P} = 0\vec{P} }$, which effectively cancels out the ${ \vec{P} }$ component. Thus, the resultant vector ${ \vec{R} }$ simplifies to: ${ \vec{R} = 4\vec{Q} }$ This resultant vector is a crucial intermediate step in our solution, as it represents the combined effect of the original two vectors. With the resultant vector now in hand, we can proceed to the next phase of the problem: determining the angle between this resultant and the vector ${ \vec{Q} }$. This will involve using the dot product formula, which provides a means to relate the magnitudes of the vectors and the cosine of the angle between them. The simplified form of the resultant vector, ${ \vec{R} = 4\vec{Q} }$, makes the subsequent calculations more straightforward, highlighting the importance of simplifying vector expressions whenever possible.

With the resultant vector ${ \vec{R} = 4\vec{Q} }$ determined, the next crucial step is to calculate the angle between ${ \vec{Q} }$ and ${ \vec{R} }$. To achieve this, we employ the dot product formula, a cornerstone of vector algebra. The dot product, denoted by the symbol '⋅', relates two vectors' magnitudes and the cosine of the angle between them. Mathematically, the dot product of vectors ${ \vec{A} }$ and ${ \vec{B} }$ is defined as: ${ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta) }$ where ${ |\vec{A}| }$ and ${ |\vec{B}| }$ represent the magnitudes of vectors ${ \vec{A} }$ and ${ \vec{B} }$ respectively, and ${ \theta }$ is the angle between them. In our case, we want to find the angle between ${ \vec{Q} }$ and ${ \vec{R} }$, so we can write: ${ \vec{Q} \cdot \vec{R} = |\vec{Q}| |\vec{R}| \cos(\theta) }$ Substituting ${ \vec{R} = 4\vec{Q} }$ into the equation, we get: ${ \vec{Q} \cdot (4\vec{Q}) = |\vec{Q}| |4\vec{Q}| \cos(\theta) }$ The dot product of ${ \vec{Q} }$ and ${ 4\vec{Q} }$ can be simplified as: ${ 4(\vec{Q} \cdot \vec{Q}) = 4|\vec{Q}|^2 }$ And the magnitudes can be written as: ${ |\vec{Q}| |4\vec{Q}| = 4|\vec{Q}|^2 }$ Thus, the equation becomes: ${ 4|\vec{Q}|^2 = 4|\vec{Q}|^2 \cos(\theta) }$ To solve for ${ \cos(\theta) }$, we divide both sides by ${ 4|\vec{Q}|^2 }$: ${ \cos(\theta) = 1 }$ This result indicates that the cosine of the angle between the vectors is 1, which is a significant clue for determining the angle itself.

Having determined that ${ \cos(\theta) = 1 }$, the final step in solving the problem is to find the angle ${ \theta }$ itself. The cosine function equals 1 at specific angles, and in the context of vector angles, we are generally looking for solutions within the range of 0 to 180 degrees (or 0 to ${ \pi }$ radians). The angle whose cosine is 1 is 0 degrees. This can be expressed mathematically as: ${ \theta = \cos^{-1}(1) }$ ${ \theta = 0^\circ }$ This result implies that the vector ${ \vec{Q} }$ and the resultant vector ${ \vec{R} }$ are parallel and point in the same direction. This conclusion aligns with our earlier finding that ${ \vec{R} = 4\vec{Q} }$, which means ${ \vec{R} }$ is simply a scalar multiple of ${ \vec{Q} }$. When two vectors are scalar multiples of each other, they lie on the same line and hence the angle between them is either 0 degrees or 180 degrees. In this specific case, since the scalar is positive (4), the vectors point in the same direction, resulting in an angle of 0 degrees. Therefore, the angle between the vector ${ \vec{Q} }$ and the resultant of ${ (2\vec{Q} + 2\vec{P}) }$ and ${ (2\vec{Q} - 2\vec{P}) }$ is 0 degrees. This completes the solution to the problem, demonstrating the application of vector addition, the dot product, and trigonometric principles to find the angle between vectors.

In summary, the problem required us to determine the angle between a vector ${ \vec{Q} }$ and the resultant of two vector expressions, ${ (2\vec{Q} + 2\vec{P}) }$ and ${ (2\vec{Q} - 2\vec{P}) }$. By systematically applying vector algebra principles, we first calculated the resultant vector ${ \vec{R} }$, which simplified to ${ 4\vec{Q} }$. Subsequently, we employed the dot product formula to relate the vectors and the cosine of the angle between them. This led us to the equation ${ \cos(\theta) = 1 }$, which indicated that the angle ${ \theta }$ is 0 degrees. This result signifies that the vectors ${ \vec{Q} }$ and ${ \vec{R} }$ are parallel and point in the same direction, reinforcing the understanding that scalar multiples of a vector lie along the same line. The problem effectively illustrates the interconnectedness of vector addition, the dot product, and trigonometric functions in solving vector-related problems. The methodology used here can be extended to a wide range of vector problems, making it a valuable tool in various fields, including physics, engineering, and computer graphics. Understanding these concepts is crucial for anyone working with vector quantities and their applications. This exercise not only provided a solution to a specific problem but also deepened our comprehension of the fundamental principles of vector mathematics.

Answer: The angle between vector ${ \vec{Q} }$ and the resultant of ${ (2\vec{Q} + 2\vec{P}) }$ and ${ (2\vec{Q} - 2\vec{P}) }$ is ${ 0^\circ }$.