91Ó°ÊÓ

In Problems, find an equation of the plane that satisfies the given conditions. Contains \((2,4,8)\) and is perpendicular to the line \(x=10-3 t\), \(y=5+t, z=6-\frac{1}{2} t\)

In Problems, find an equation of the plane that satisfies the given conditions. Contains \((1,1,1)\) and is perpendicular to the line through \((2,6,-3)\) and \((1,0,-2)\)

Prove or disprove \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}\).

Let \(\mathscr{P}_{1}\) and \(\mathscr{P}_{2}\) be planes with normal vectors \(\mathbf{n}_{1}\) and \(\mathbf{n}_{2}\), respectively. \(\mathscr{P}_{1}\) and \(\Phi_{2}\) are orthogonal if \(\mathbf{n}_{1}\) and \(\mathbf{n}_{2}\) are orthogonal and parallel if \(\mathbf{n}_{1}\) and \(\mathbf{n}_{2}\) are parallel. Determine which of the following planes are orthogonal and which are parallel. (a) \(2 x-y+3 z=1\) (b) \(x+2 y+2 z=9\) (c) \(x+y-\frac{3}{2} z=2\) (d) \(-5 x+2 y+4 z=0\) (e) \(-8 x-8 y+12 z=1\) (f) \(-2 x+y-3 z=5\)

Prove \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})=(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}\)

Find parametric equations for the line that contains \((-4,1,7)\) and is perpendicular to the plane \(-7 x+2 y+3 z=1\).

Prove \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})+\mathbf{b} \times(\mathbf{c} \times \mathbf{a})+\mathbf{c} \times(\mathbf{a} \times \mathbf{b})=\mathbf{0}\)

Determine which of the following planes are perpendicular to the line \(x=4-6 t, y=1+9 t, z=2+3 t\) (a) \(4 x+y+2 z=1\) (b) \(2 x-3 y+z=4\) (c) \(10 x-15 y-5 z=2\) (d) \(-4 x+6 y+2 z=9\)

Prove Lagrange's identity: $$ \|\mathbf{a} \times \mathbf{b}\|^{2}=\|\mathbf{a}\|^{2}\|\mathbf{b}\|^{2}-(\mathbf{a} \cdot \mathbf{b})^{2} . $$

Determine which of the following planes are parallel to the line \((1-x) / 2=(y+2) / 4=z-5\) (a) \(x-y+3 z=1\) (b) \(6 x-3 y=1\) (c) \(x-2 y+5 z=0\) (d) \(-2 x+y-2 z=7\)