Formulas (1), (2), (3), (5), and (10), which apply toplanes in 3-space, have analogs for

The point-normal form of the equation of the plane through (0, 3, 5) and perpendicular to \(\langle-4,1,7\rangle\) is ________. Equation Transcription: ? -4, 1, 7? Text Transcription: left angle -4, 1, 7 right angle

A normal vector for the plane \(4 x-2 y+7 z-11=0\) is _______. Equation Transcription: Text Transcription: 4x - 2y + 7z -11 = 0

A normal vector for the plane through the points (2, 5, 1), (3, 7, 0), and (2, 5, 2) is .

The acute angle of intersection of the planes \(x+y-2 z=5\) and \(3 y-4 z=6\) is _______. Equation Transcription: Text Transcription: x + y - 2z = 5 3y - 4z = 6

The distance between the point (9, 8, 3) and the plane \(x+y-2 z=5\) is _______. Equation Transcription: Text Transcription: x + y - 2z = 5

Find an equation of the plane that passes through the point \(P\) and has the vector \(n\) as a normal. \(P(0,0,0) ; n=\langle 2,-3,-4\rangle\) Equation Transcription: ?2, -3, -4? Text Transcription: P n P (0, 0, 0); n = left angle 2, -3, -4 right angle

710 Find an equation of the plane indicated in the figure. 7

710 Find an equation of the plane indicated in the figure. 8

710 Find an equation of the plane indicated in the figure. 9

710 Find an equation of the plane indicated in the figure. 10

1112 Find an equation of the plane that passes through the given points. (2, 1, 1), (0, 2, 3), and (1, 0, 1)

1112 Find an equation of the plane that passes through the given points. (3, 2, 1), (2, 1, 1), and (1, 3, 2)

Determine whether the planes are parallel, perpendicular, or neither. (a) \(2 x-8 y-6 z-2=0\) \(-x+4 y+3 z-5=0\) (b) \(3 x-2 y+z=1\) \( 4 x+5 y-2 z=4\) (c) \(x-y+3 z-2=0\) Equation Transcription: Text Transcription: 2x - 8y - 6z -2 = 0 -x + 4y + 3z - 5 = 0 3x - 2y + z = 1 4x + 5y - 2z = 4 x - y + 3z - 2 = 0

Determine whether the planes are parallel, perpendicular, or neither. (a) \(3 x-2 y+z=4\) \(6 x-4 y+3 z=7\) (b) \(y=4 x-2 z+3=1\) \(x=\frac{1}{4} y+\frac{1}{2} z\) (c) \(x+4 y+7 z=3\) \(5 x-3 y+z=0\) Equation Transcription: Text Transcription: 3 x -2y + z = 4 6x - 4y + 3z = 7 y = 4x - 2z + 3 = 1 x = frac{1}{4} y + frac{1}{2} z x + 4y + 7z = 3 5x - 3 y + z = 0

Determine whether the line and plane are parallel, perpendicular, or neither. (a) \(x=4+2 t, y=-t, z=-1-4 t\) \(3 x+2 y+z-7=0\) (b) \(x=t, y=2 t, z=3 t\) \(x-y+2 z=5\) (c) \(x=-1+2 t, y=4+t, z=1-t\) \(4 x+2 y-2 z=7\) Equation Transcription: Text Transcription: x=4+2 t, y=-t, z=-1-4 t 3 x+2 y+z-7=0 x=t, y=2 t, z=3 t x-y+2 z=5 x=-1+2 t, y=4+t, z=1-t 4 x+2 y-2 z=7

Determine whether the line and plane are parallel, perpendicular, or neither. (a) \(x=3-t, y=2+t, z=1-4 t\) \(2 x+2 y-5=0\) (b) \(x=1-2 t, y=t, z=-t;\) \(6 x-3 y+3 z=1\) (c) \(x=t, y=1-t, z=2+t\) \(x+y+z=1\) Equation Transcription: Text Transcription: x = 3 - t, y = 2 + t, z = 1 - 4t 2x + 2y - 5 = 0 x = 1 - 2t, y = t, z = -t 6x - 3y + 3z = 1 x = t, y = 1 - t, z = 2 + t x + y + z = 1

Determine whether the line and plane intersect; if so, find the coordinates of the intersection. (a) \(x=t, y=t, z=t\) \(3 x-2 y+z-5=0\) (b) \(x=2-t, y=3+t, z=t\) \(2 x+y+z=1\) Equation Transcription: Text Transcription: x = t, y = t, z = t 3x - 2y + z - 5 = 0 x = 2 - t, y = 3 + t, z = t 2x + y + z = 1

Determine whether the line and plane intersect; if so, find the coordinates of the intersection. (a) \(x=3 t, y=5 t, z=-t\); \(2 x-y+z+1=0\) (b) \(x=1+t, y=-1+3 t, z=2+4 t\) \(x-y+4 z=7\) Equation Transcription: Text Transcription: x = 3 t, y = 5 t, z = -t; 2x - y + z + 1 = 0 x = 1 + t, y = -1 + 3t, z = 2 + 4t x - y + 4z = 7

Find the acute angle of intersection of the planes to the nearest degree. \( x=0 \text { and } 2 x-y+z-4=0\) Equation Transcription: and Text Transcription: x = 0 and 2x - y + z - 4 = 0

Find the acute angle of intersection of the planes to the nearest degree. \(x+2 y-2 z=5 \text { and } 6 x-3 y+2 z=8\) Equation Transcription: and Text Transcription: X + 2y - 2z = 5 and 6x - 3y + 2z = 8

2124 TrueFalse Determine whether the statement is true or false. Explain your answer. Every plane has exactly two unit normal vectors.

2124 TrueFalse Determine whether the statement is true or false. Explain your answer. If a plane is parallel to one of the coordinate planes, then its normal vector is parallel to one of the three vectors i, j, or k.

True-False Determine whether the statement is true or false. Explain your answer. If two planes intersect in a line \(L\), then \(L\) is parallel to the cross product of the normals to the two planes. Equation Transcription: Text Transcription: L L

True-False Determine whether the statement is true or false. Explain your answer. If \(a^{2}+b^{2}+c^{2}=1\), then the distance from \(P\left(x_{0}, y_{0}, z_{0}\right)\) to the plane \(a x+b y+c z=0\) is \(\left|\langle a, b, c\rangle \cdot\left\langle x_{0}, y_{0,} z_{0}\right\rangle\right|\) Equation Transcription: |?????| Text Transcription: a^2 + b^2 +c^2 = 1 P(x_0, y_0, z_0) ax + by + cz = 0 |left angle a, b, c right angle . left angle x_0, y_0, z_0 right angle|

Find an equation of the plane that satisfies the stated conditions. The plane through the origin that is parallel to the plane \(4 x-2 y+7 z+12=0\) Equation Transcription: Text Transcription: 4 x-2 y+7 z+12=0

Find an equation of the plane that satisfies the stated conditions. The plane that contains the line \(x=-2+3 t, y=4+2 t, z=3-t\) and is perpendicular to the plane \(x-2 y+z=5\) Equation Transcription: Text Transcription: x = -2 + 3t, y = 4 + 2t, z = 3 - t x - 2y + z = 5

Find an equation of the plane that satisfies the stated conditions. The plane through the point (-1, 4, 2) that contains the line of intersection of the planes \(4 x-y+z-2=0\) and \(2 x+y-2 z-3=0\). Equation Transcription: Text Transcription: 2x + y - 2z - 3 = 0 4x - y + z - 2 = 0

Find an equation of the plane that satisfies the stated conditions. The plane through (-1, 4, _3) that is perpendicular to the line \(x-2=t, y+3=2 t, z=-t\). Equation Transcription: Text Transcription: x - 2 = t, y + 3 = 2t, z = -t

Find an equation of the plane that satisfies the stated conditions. The plane through (1, 2, -1) that is perpendicular to the line of intersection of the planes \(2 x+y+z=2\) and \(x+2 y+z=3\) Equation Transcription: Text Transcription: 2x + y + z = 2 x + 2y + z = 3

Find an equation of the plane that satisfies the stated conditions. The plane through the points \(P_{1}(-2,1,4), P_{2}(1,0,3)\) that is perpendicular to the plane \(4 x-y+3 z=2\) Equation Transcription: Text Transcription: P_1(-2,1,4), P_2(1,0,3 4x - y + 3z = 2

Find an equation of the plane that satisfies the stated conditions. The plane through (-1, 2, -5) that is perpendicular to the planes \(2 x-y+z=1\) and \(x+y-2 z=3\). Equation Transcription: Text Transcription: 2x - y + z = 1 x + y - 2z = 3

Find an equation of the plane that satisfies the stated conditions. The plane that contains the point (2, 0, 3) and the line \(x=-1+t, y=t, z=-4+2 t\). Equation Transcription: Text Transcription: x = -1 + t, y = t, z = -4 + 2t

2534 Find an equation of the plane that satisfies the stated conditions. The plane whose points are equidistant from (2, 1, 1) and (3, 1, 5).

Find an equation of the plane that satisfies the stated conditions. The plane that contains the line \(x=3 t, y=1+t, z=2 t\) and is parallel to the intersection of the planes \(y+z=-1\) and \(2 x-y+z+0\). Equation Transcription: Text Transcription: x = 3t, y = 1 + t, z = 2t y + z = -1 2x - y + z + 0

Find parametric equations of the line through the point (5, 0, -2) that is parallel to the planes \(x-4 y+2 z=0\) and \(2 x+3 y-z+1=0\) Equation Transcription: Text Transcription: x - 4y + 2z = 0 2x + 3y - z + 1 = 0

Let \(L\) be the line \(x=3 t+1, y=-5 t, z=t\) (a) Show that \(L\) lies in the plane \(2 x+y-z=2\) (b) Show that \(L\) is parallel to the plane \(x+y+2 z=0\). Is the line above, below, or on this plane? Equation Transcription: Text Transcription: L x = 3t + 1, y = -5t, z = t L 2x + y - z = 2 L x + y + 2z = 0

Show that the lines \(x=-2+t, y=3+2 t, z=4-t\) \(x=3-t, y=4-2 t, z=t\) are parallel and find an equation of the plane they determine. Equation Transcription: Text Transcription: x = -2 + t, y = 3 + 2t, z = 4 - t x = 3t, y = 4 - 2t, z = t

Show that the lines \(L_{1}: x+1=4 t, y-3=t, z-1=0\) \(L_{2}: x+13=12 t, y-1=6 t, z-2=3 t\) intersect and find an equation of the plane they determine. Equation Transcription: Text Transcription: L_1: x + 1 = 4t, y - 3 = t, z - 1 = 0 L_2: x + 13 = 12t, y - 1 = 6t, z - 2 = 3t

Do the points (1, 0, 1), (0, 2, 3), (2, 1, 1), and (4, 2, 3) lie in the same plane? Justify your answer two different ways.

Focus on Concepts Show that if \(a, b,\) and \(c\) are nonzero, then the plane whose intercepts with the coordinate axes are \(x=a\), \(y=b\), and \(z=c\)is given by the equation \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\) Equation Transcription: and and Text Transcription: a, b, and c x = a y = b, and z = c frac{x}{a} + frac{y}{b} + frac{z}{c} = 1

Find parametric equations of the line of intersection of the planes. \(-2 x+3 y+7 z+2=0\) \(x+2 y-3 z+5=0\) Equation Transcription: Text Transcription: -2x + 3y + 7z + 2 = 0 x + 2y - 3z + 5 = 0

Find parametric equations of the line of intersection of the planes. \(2 x-5 y+2 z=0\) \(z=0\) Equation Transcription: Text Transcription: 2x - 5y + 2z = 0 z = 0

Find the distance between the point and the plane. \((1,-2,3) ; 2 x-2 y+z=4\) Equation Transcription: Text Transcription: (1,-2,3); 2x - 2y + z = 4

Find the distance between the point and the plane. \((0,1,5) ; \quad 3 x+6 y-2 z-5=0\) Equation Transcription: Text Transcription: (0,1,5) ; 3x + 6y - 2z - 5 = 0

Find the distance between the given parallel planes. \(-2 x+y+z=0\) \(6 x-3 y-3 z-5=0\) Equation Transcription: Text Transcription: -2x + y + z = 0 6x - 3y - 3z - 5 = 0

Find the distance between the given parallel planes. \(x+y+z=1\) \(x+y+z=-1\) Equation Transcription: Text Transcription: x + y + z = 1 x + y + z = -1

Find the distance between the given skew lines. \(x=1+7 t, y=3+t, z=5-3 t\) \(x=4-t, y=6, z=7+2 t\) Equation Transcription: Text Transcription: x = 1 + 7t, y = 3 + t, z = 5 - 3t x = 4 - t, y = 6, z = 7 + 2t

Find the distance between the given skew lines. \(x=3-t, y=4+4 t, z=1+2 t\) \(x=t, y=3, z=2 t\) Equation Transcription: Text Transcription: x = 3 - t, y = 4 + 4t, z = 1 + 2t x = t, y = 3, z = 2t

Find an equation of the sphere with center (2, 1, -3) that is tangent to the plane \(x-3 y+2 z=4\). Equation Transcription: Text Transcription: x - 3y + 2z = 4

Locate the point of intersection of the plane \(2 x+y-z=0\) and the line through (3, 1, 0) that is perpendicular to the plane. Equation Transcription: Text Transcription: 2x + y - z = 0

Show that the line \(x=-1+t, y=3+2 t, z=-t\) and the plane \(2 x-2 y-2 z+3=0\) are parallel, and find the distance between them. Equation Transcription: Text Transcription: x = -1 + t, y = 3 + 2t, z = -t 2x - 2y - 2z + 3 = 0

Focus on Concepts Formulas (1), (2), (3), (5), and (10), which apply to planes in 3 - space, have analogs for lines in 2-space. (a) Draw an analog of Figure 11.6.3 in 2-space to illustrate that the equation of the line that passes through \(P\left(x_{0}, y_{0}\right)\) and is perpendicular to the vector \(n=\langle a, b\rangle\) can be expressed as \(n \cdot\left(r-r_{0}\right)=0\) where \(r=\langle x, y\rangle\) and \(r_{0}=\left\langle x_{0}, y_{0}\right\rangle\). (b) Show that the vector equation in part (a) can be expressed as \(a\left(x-x_{0}\right)+b\left(y-y_{0}\right)=0\) This is called the point-normal form of a line. (c) Using the proof of Theorem 11.6.1 as a guide, show that if \(a\) and \(b\) are not both zero, then the graph of the equation \(a x+b y+c=0\) is a line that has \(n=\langle a, b\rangle\) as a normal. (d) Using the proof of Theorem 11.6.2 as a guide, show that the distance \(D\) between a point \(P\left(x_{0}, y_{0}\right)\) and the line \(a x+b y+c=0\) is \(D=\frac{\left|a x_{0}+b y_{0}+c\right|}{\sqrt{a^{2}+b^{2}}}\) (e) Use the formula in part (d) to find the distance between the point \(P(-3,5)\) and the line \(y=-2 x+1\). Equation Transcription: ?? ?? ?? ?? Text Transcription: P(x_0, y_0) n = left angle a, b right angle n cdot (r-r_0) = 0 r = left angle x, y right angle r_0 =left angle x_0, y_0 right angle. a(x - x_0) + b(y - y_0) = 0 a b ax + by + c = 0 n = left angle a, b right angle D P(x_0, y_0) ax + by + c = 0 D = frac{|ax_0 +by_0 + c|}{sqrt{a^2 +b^2} P(-3, 5) y = -2x + 1

Focus on Concepts (a) Show that the distance \(D\) between parallel planes \(a x+b y+c z+d_{1}=0\) \(a x+b y+c z+d_{2}=0\) is \(D=\frac{\left|d_{1}-d_{2}\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}\) (b) Use the formula in part (a) to solve Exercise 45 . Equation Transcription: Text Transcription: D ax + by + cz + d_1 = 0 ax + by + cz + d_2 = 0 D = frac{|d_1 - d_2|}{sqrt{a^2 + b^2 + c^2}

Writing Explain why any line in 3-space must lie in some vertical plane. Must any line in 3-space also lie in some horizontal plane?

Writing Given two planes, discuss the various possibilities for the set of points they have in common. Then consider the set of points that three planes can have in common.