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Chapter 3: Q23P (page 112)

Find the angle between the given planes.
$2 x + y - 2 z = 3$ and $3 x - 6 y - 2 z = 4$

Short Answer

Expert verified

The angle between the planes is $θ \approx 79$ .

Step by step solution

01

Concept and formula used

The angle formed by the planes and the normal to the planes is same.

To determine the angle between the vectors by using the formula $θ = c o s^{- 1} \frac{A \times B}{| A | | B |}$ .

02

Find the angle

The given planes are, $2 x + y - 2 z = 3$ and $3 x - 6 y - 2 z = 4$ .

Calculating gives the values as below:

$A = |A| = \sqrt{{(2)}^{2} + {(1)}^{2} + {(- 2)}^{2}} = 3 B = |B| = \sqrt{{(3)}^{2} + {(- 6)}^{2} + {(- 2)}^{2}} = 7 A \times B = (2) (3) + (1) (- 6) + (- 2) (- 2) = 6 - 6 + 4 = 4$

Now find the angle:

$θ = c o s^{- 1} \frac{A \times B}{A B} = c o s^{- 1} \frac{4}{7 \times 3} = c o s^{- 1} \frac{4}{21} \approx 79$

The angle between the planes is $θ \approx 79$ .

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Most popular questions from this chapter

As in Problem 1, write out in detail in terms of equations like (2.6) for two equations in four unknowns; for four equations in two unknowns.

Find the symmetric equations $(5.6) or 5.7)$ and the parametric equations $(5.8)$ of a line, and/or the equation $(5.10)$ of the plane satisfying the following given conditions.

Line through $(5, - 4, 2)$ and parallel to the line $r = i - j + (5 i - 2 j + k) t .$

Show that each of the following matrices is orthogonal and find the rotation and/or reflection it produces as an operator acting on vectors. If a rotation, find the axis and angle; if a reflection, find the reflecting plane and the rotation, if any, about the normal to that plane.

$M = \frac{1}{2} (\begin{matrix} 1 & \sqrt{2} & - 1 \\ \sqrt{2} & 0 & \sqrt{2} \\ 1 & - \sqrt{2} & - 1 \end{matrix})$

(a): As in problem 12,
$D^{2} + 2 D + 1$ linear?

(b): Is $x^{2} D^{2} - 2 x D + 7$ a linear operator?

For each of the following problems write and row reduce the augmented matrix to find out whether the given set of equations has exactly one solution, no solutions, or an infinite set of solutions. Check your results by computer. Warning hint:Be sure your equations are written in standard form. Comment: Remember that the point of doing these problems is not just to get an answer (which your computer will give you), but to become familiar with the terminology, ideas, and notation we are using.

3.

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