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Chapter 3: Q21P (page 112)

In Problem 21 to 23, find the angle between the given planes.
$2 x + 6 y - 3 z = 10$ and $5 x + 2 y - z = 12$

Short Answer

Expert verified

The angle between the given points is $θ = 49.3$ .

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, $2 x + 6 y - 3 z = 10$ and $5 x + 2 y - z = 12$ .

Calculating gives the values as below:

$A = |A| = \sqrt{{(2)}^{2} + {(6)}^{2} + {(- 3)}^{2}} = 7 B = |B| = \sqrt{{(5)}^{2} + {(2)}^{2} + {(- 1)}^{2}} = \sqrt{30} A \times B = (2) (5) + (6) (2) + (- 3) (- 1) = 10 + 12 + 3 = 25$

Now find the angle:

$θ = c o s^{- 1} \frac{25}{7 \times \sqrt{30}} = 49.3$

Thus, the required angle is $49.3$ .

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Most popular questions from this chapter

Show that the definition of a Hermitian matrix $(A = A^{t})$ can be writtenrole="math" localid="1658814044380" $a_{ij} = {\bar{a}}_{ji}$ (that is, the diagonal elements are real and the other elements have the property that $a_{12} = {\bar{a}}_{21}$ , etc.). Construct an example of a Hermitian matrix.

Find the symmetric equations $(5.6) o r (5.7)$ 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 $(4, - 1, 3)$ and parallel to $i - 2 k$ .

In Problems $8 to 15, use (8.5)$ to show that the given functions are linearly independent.

$13. \sin x, \sin 2 x$

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.

$6 . \{\begin{cases} x + y - z = 1 \\ 3 x + 2 y - 2 z = 3 \end{cases}$

The diagonals of a rhombus (four-sided figure with all sides of equal length) are perpendicular and bisect each other.

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