Question
Determine if the given planes are parallel, perpendicular, or neither. \( \begin{array}{l}-2 x+2 y-4 z+5=0 \\ 4 x-4 y+8 z-5=0 \\ \checkmark \text { Select } \\ \text { Parallel } \\ \text { Neither } \\ \text { Perpendicular }\end{array} \)
Ask by Goodwin Best. in the United States
Mar 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The planes are parallel.
Solution
We identify the normal vectors of each plane.
The first plane is
\[
-2x + 2y - 4z + 5 = 0,
\]
so its normal vector is
\[
\mathbf{n}_1 = \langle -2,\, 2,\, -4 \rangle.
\]
The second plane is
\[
4x - 4y + 8z - 5 = 0,
\]
so its normal vector is
\[
\mathbf{n}_2 = \langle 4,\, -4,\, 8 \rangle.
\]
To determine if the planes are parallel, we check whether the normal vectors are scalar multiples of each other. Notice that
\[
\mathbf{n}_2 = -2 \cdot \langle -2,\, 2,\, -4 \rangle = \langle 4,\, -4,\, 8 \rangle.
\]
Since the normal vector \(\mathbf{n}_2\) is a scalar multiple of \(\mathbf{n}_1\), the planes are parallel.
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Simplify this solution Bonus Knowledge
To determine the relationship between the two planes, we need to look at their normal vectors. The normal vector of the first plane \( -2x + 2y - 4z + 5 = 0 \) is given by \((-2, 2, -4)\), and the normal vector of the second plane \( 4x - 4y + 8z - 5 = 0 \) is \((4, -4, 8)\). First, check for parallelism: Two planes are parallel if their normal vectors are scalar multiples of each other. In this case, the second normal vector \((4, -4, 8)\) is indeed a scalar multiple of the first one \((-2, 2, -4)\) since \(4 = -2 \times (-2)\), \(-4 = 2 \times (-2)\), and \(8 = -4 \times (-2)\). This indicates that the planes are parallel. Therefore, the answer is: **Parallel**
