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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

Find AB, BA , A+B , A-B , $A^{2}$ , $B^{2}$ ,5.A,3,B . Observe that $AB \neq BA$ . Show that $(A - B) (A + B) \neq (A + B) (A - B) \neq A^{2} - B^{2}$ . Show that $\det (AB) = \det (BA) = (detA) (detB)$ , but that $\det (A + B) \neq detA + detB$ . Show that $\det (5 A) \neq 5 detA$ and find n so that $\det (5 A) = 5^{n} detA$ . Find similar results for $\det (3 B)$ . Remember that the point of doing these simple problems by hand is to learn how to manipulate determinants and matrices correctly. Check your answers by computer.

role="math" localid="1658986967380" $A = (\begin{matrix} 1 & 0 & 2 \\ 3 & - 1 & 0 \\ 0 & 5 & 1 \end{matrix}), B = (\begin{matrix} 1 & 1 & 0 \\ 0 & 2 & 1 \\ 3 & - 1 & 0 \end{matrix})$

Find the distance between the two given lines.

$\frac{x - 1}{2} = \frac{y + 2}{3} = \frac{2 z - 1}{4} and \frac{x + 2}{- 1} = \frac{2 - y}{2}, z = \frac{1}{2}$

Question: Find the values of $λ$ such that the following equations have nontrivial solutions, and for each $λ$ , solve the equations.

Find the rank of each of the following matrices.

$(\begin{matrix} 1 & 1 & 2 \\ 2 & 4 & 6 \\ 3 & 2 & 5 \end{matrix})$

Let each of the following matrices represent an active transformation of vectors in (x,y)plane (axes fixed, vector rotated or reflected). As in Example 3, show that each matrix is orthogonal, find its determinant and find its rotation angle, or find the line of reflection. $\frac{1}{5} (\begin{matrix} 3 & 4 \\ 4 & - 3 \end{matrix})$ $\frac{1}{5} (\begin{matrix} 3 & 4 \\ 4 & - 3 \end{matrix})$

See all solutions

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