Question

Determine whether the planes $a_{1} x+b_{1} y+c_{1} z=d_{1}$ and $a_{2} x+b_{2} y+c_{2} z=d_{2}$ are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant $k$ such that $a_{1}=k a_{2}, b_{1}=k b_{2}$, and $c_{1}=k c_{2},$ and are perpendicular if $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$. $$ 3 x+y-4 z=3,-9 x-3 y+12 z=4 $$

Name: Problem 22, Chapter 7: Calculus an Applied Approach
Uploaded: 2020-11-04T15:44:44-08:00
Duration: 1 min 23 s
Channel: Linh Vu
Description: Determine whether the planes a1 x+b1 y+c1 z=d1 and a2 x+b2 y+c2 z=d2 are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant k such that a1=k a2, b1=k b2, and c1=k c2, and are perpendicular if a1 a2+b1 b2+c1 c2=0. 3 x+y-4 z=3,-9 x-3 y+12 z=4

   Determine whether the planes $a_{1} x+b_{1} y+c_{1} z=d_{1}$ and $a_{2} x+b_{2} y+c_{2} z=d_{2}$ are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant $k$ such that $a_{1}=k a_{2}, b_{1}=k b_{2}$, and $c_{1}=k c_{2},$ and are perpendicular if $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$.
$$
3 x+y-4 z=3,-9 x-3 y+12 z=4
$$

Calculus an Applied Approach

Ron Larson, David C,… 8th Edition

Chapter 7, Problem 22

Instant Answer

Solved by Expert Linh Vu

11/04/2020

Step 1

The normal vector of a plane is given by the coefficients of x, y, and z in the equation of the plane. For the first plane, $3x + y - 4z = 3$, the normal vector $\vec{n_1}$ is $(3, 1, -4)$. For the second plane, $-9x - 3y + 12z = 4$, the normal vector Show more…

Show all steps

Thanks for your feedback!

Determine whether the planes $a_{1} x+b_{1} y+c_{1} z=d_{1}$ and $a_{2} x+b_{2} y+c_{2} z=d_{2}$ are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant $k$ such that $a_{1}=k a_{2}, b_{1}=k b_{2}$, and $c_{1}=k c_{2},$ and are perpendicular if $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$. $$ 3 x+y-4 z=3,-9 x-3 y+12 z=4 $$

Play audio

Feedback

Linh Vu and 77 other subject Calculus 1 / AB educators are ready to help you.

Ask a new question

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Normal Vector

A plane given by an equation of the form ax + by + cz = d has a normal vector (a, b, c). This vector is perpendicular to every line lying on the plane and is key to understanding a plane’s orientation in space.

Parallelism of Planes

Two planes are parallel if their normal vectors are parallel. This means that one normal vector is a scalar multiple of the other, indicating that their orientations are identical even if their positions are different.

Perpendicularity of Planes

Two planes are perpendicular if their normal vectors are perpendicular. This is determined by the dot product of the normal vectors being zero, which means the vectors form a right angle.

Dot Product

The dot product of two vectors is a scalar value that reflects the degree of their alignment. In the context of planes, if the dot product of their normal vectors is zero, the planes are perpendicular; if one is a scalar multiple of the other, the planes are parallel.

Please give Ace some feedback