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If the vector $P = (1,2,-1)$ is parallel to the plane $7x+2y+kz = 5$, then what's the value of $k$?
Finding Points Parallel to a Plane
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SUMMARY
The discussion centers on determining the value of \( k \) for the plane equation \( 7x + 2y + kz = 5 \) such that the vector \( P = (1, 2, -1) \) is parallel to the plane. The solution involves using the normal vector \( \mathbf{n} = (7, 2, k) \) and applying the formula \( \mathbf{n} \cdot (\mathbf{x - x_0}) = 0 \). Through calculations, it is established that \( k = 6 \) to maintain parallelism with the given point.
PREREQUISITES- Understanding of vector mathematics and normal vectors
- Familiarity with plane equations in three-dimensional space
- Knowledge of the dot product and its geometric interpretation
- Basic algebra for solving equations
- Study the properties of normal vectors in vector geometry
- Learn how to derive equations of planes given points and normal vectors
- Explore the concept of parallelism in three-dimensional geometry
- Investigate applications of planes and vectors in physics and engineering
Students and professionals in mathematics, physics, and engineering who are working with three-dimensional geometry and vector analysis will benefit from this discussion.
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