Express \( \vec{V}_1 \cdot \vec{V}_1 \) in terms of \( V_1 \). If \( \vec{V}_1 \) and \( \vec{V}_2 \) are perpendicular, find \( \vec{V}_1 \cdot \vec{V}_2 \). If \( \vec{V}_1 \) an... Express \( \vec{V}_1 \cdot \vec{V}_1 \) in terms of \( V_1 \). If \( \vec{V}_1 \) and \( \vec{V}_2 \) are perpendicular, find \( \vec{V}_1 \cdot \vec{V}_2 \). If \( \vec{V}_1 \) and \( \vec{V}_2 \) are parallel, express \( \vec{V}_1 \cdot \vec{V}_2 \) in terms of \( V_1 \) and \( V_2 \).
Understand the Problem
The question relates to vector mathematics, specifically calculating the dot products of the vectors \( \vec{V}_1 \) and \( \vec{V}_2 \) under different conditions (part F, G, and H). It requires finding the expressions for these dot products based on whether the vectors are parallel or perpendicular and expressing the answers in terms of their lengths.
Answer
Part F: \( V_1^2 \) Part G: \( 0 \) Part H: \( V_1 V_2 \)
Answer for screen readers
Part F: ( \vec{V}_1 \cdot \vec{V}_1 = V_1^2 )
Part G: ( \vec{V}_1 \cdot \vec{V}_2 = 0 )
Part H: ( \vec{V}_1 \cdot \vec{V}_2 = V_1 V_2 )
Steps to Solve
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Dot Product Definition The dot product of two vectors, $\vec{A}$ and $\vec{B}$, is given by the formula: $$ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta) $$ where $|\vec{A}|$ and $|\vec{B}|$ are the magnitudes of the vectors, and $\theta$ is the angle between them.
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Part F: Finding $\vec{V}_1 \cdot \vec{V}_1$ Given that we're calculating the dot product of a vector with itself, we have: $$ \vec{V}_1 \cdot \vec{V}_1 = |\vec{V}_1|^2 $$ Thus, $|\vec{V}_1|$ is simply the length of $\vec{V}_1$ (denoted as $V_1$), so: $$ \vec{V}_1 \cdot \vec{V}_1 = V_1^2 $$
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Part G: Finding $\vec{V}_1 \cdot \vec{V}_2$ when perpendicular Since $\vec{V}_1$ and $\vec{V}_2$ are perpendicular, the angle $\theta$ between them is $90^\circ$. Therefore, $\cos(90^\circ) = 0$: $$ \vec{V}_1 \cdot \vec{V}_2 = |\vec{V}_1| |\vec{V}_2| \cdot 0 = 0 $$
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Part H: Finding $\vec{V}_1 \cdot \vec{V}_2$ when parallel When $\vec{V}_1$ and $\vec{V}_2$ are parallel, the angle $\theta$ between them is $0^\circ$. Therefore, $\cos(0^\circ) = 1$: $$ \vec{V}_1 \cdot \vec{V}_2 = |\vec{V}_1| |\vec{V}_2| = V_1 V_2 $$
Part F: ( \vec{V}_1 \cdot \vec{V}_1 = V_1^2 )
Part G: ( \vec{V}_1 \cdot \vec{V}_2 = 0 )
Part H: ( \vec{V}_1 \cdot \vec{V}_2 = V_1 V_2 )
More Information
The dot product is a fundamental operation in vector mathematics, commonly used in physics and engineering to find projections and angles between vectors. The results you obtained illustrate key properties of the dot product: its dependence on the angle between vectors and how it behaves for self dot products.
Tips
- Confusing the dot product result when vectors are not aligned correctly.
- Forgetting that the dot product of a vector with itself gives the square of its magnitude. Always check that the correct case (self dot product vs. two different vectors) is being considered.
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