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Problem 581

Find the equation of the plane through the points $\mathrm{P}_{1}(1,2,-1), \mathrm{P}_{2}(2,3,1)\(, and \)\mathrm{P}_{3}(3,-1,2)$

Expert verified

The equation of the plane through the points \(P_1(1,2,-1), P_2(2,3,1)\), and \(P_3(3,-1,2)\) is \(9x + y - 5z - 16 = 0\).

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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

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

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

Show that the linear transformation \(\mathrm{T}(\mathrm{V})=\mathrm{BV}\), where \(\mathrm{B}=11 \quad 0 \mid\) \(10 \quad-1 \mid\) is the matrix of \(\mathrm{T}\) with respect to the usual basis, is an orthogonal mapping.

Chapter 23

Find the unique line passing through \((2,4)\) and \((3,-2)\) in the form $$ \mathrm{L}=\mathrm{u}+\mathrm{r}(\mathrm{v}-\mathrm{u}) $$

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