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Problem 588

Show that the points \(\mathrm{P}=(0,2,0), \mathrm{Q}=(1,2,1)\) and \(\mathrm{R}=(2,2,1)\) are noncollinear. Then find the unique plane passing through these points.

Expert verified

The points P, Q, and R are noncollinear because their direction vectors (1, 0, 1) and (2, 0, 1) are not proportional. The unique plane passing through these points has the equation \(x + y - 2 = 0\).

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Chapter 23

What are the different approaches to planes in \(\mathrm{R}^{3}\) ?

Chapter 23

Classify the rigid motions in the plane.

Chapter 23

Find a parametric representation of the line passing through \(\mathrm{P}\) and in the direction of u where (a) \(\mathrm{P}=(2,5)\) and \(\mathrm{u}=(-3,4)\); (b) \(\mathrm{P}=(4,-2,3,1)\) and \(\mathrm{u}=(2,5,-7,11)\).

Chapter 23

Find an equation of the hyperplane \(\mathrm{H}\) in the vector space \(\mathrm{R}^{4}\) if: (i) \(\mathrm{H}\) passes through \(\mathrm{P}=[3,-2,1,-4]\) and is normal to \(\mathrm{u}=[2,5,-6,-2] ;\) ii) \(\mathrm{H}\) passes through \(\mathrm{P}=[1,-2,3,5]\) and is parallel to the hyperplane \(\mathrm{H}^{\prime}\) determined by $4 \mathrm{x}-5 \mathrm{y}+2 \mathrm{z}+\mathrm{w}=11$.

Chapter 23

List the different ways in which lines can be defined in \(\mathrm{R}^{3}\).

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