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Problem 585

Describe geometrically the solutions to $\mathrm{x}_{1}-2 \mathrm{x}_{2}+4 \mathrm{x}_{3}=1$ \(3 \mathrm{x}_{1}+\mathrm{x}_{2}-\mathrm{x}_{3}=0\)

Expert verified

The given system of linear equations represents two planes in 3-dimensional space, which are not parallel. They intersect in a line with the direction vector \(\langle-2, 13, 7 \rangle\). The parametric equations of the intersection line are:
\(x = 1 - 4t + \frac{13t - 3}{3}\)
\(y = \frac{13t - 3}{6}\)
\(z = t\)
These equations describe the geometric solution to the given system of linear equations.

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Chapter 23

What is the determinantal expression for a hyperplane in i) \(\mathrm{R}^{2}\) ii) \(\mathrm{R}^{3}\) iii) \(\mathrm{R}^{\mathrm{n}_{?}}\)

Chapter 23

Find the volume of the parallelepiped determined by the vectors $u=(2,3,5), v=(-4,2,6)\( and \)\mathrm{w}=(1,0,3)\( in \)\mathrm{xyz}$ -space.

Chapter 23

Show how the set of solutions to \(2 \mathrm{x}+\mathrm{y}=4\) can be expressed as a one-dimensional subspace of \(\mathrm{R}^{2}\).

Chapter 23

Determine the nearest point in \(\mathrm{U}=\) span $\left\\{\mathrm{V}_{1}, \mathrm{~V}_{2}\right\\}\( to \)\mathrm{Y}\(, where \)\mathrm{V}_{1}=(2,1,0) \mathrm{V}_{2}=(-1,2,0) \mathrm{Y}=(1,2,3)$Geometrical Problems

Chapter 23

Classify the rigid motions in the plane.

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