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Math 126

Lecture 5

Behind us

Today: lines and planes

Wednesday: cylinders and quadric surfaces

Read Sections 12.5 and 12.6 of the book. We will not cover everything in lecture or section. Grow or die!

Questions!

Warm up

Question: how can we describe the line of intersection of two planes?

Warm up

Simpler: what is the intersection of the planes $x=0$ and $y=0$

The enemy of my enemy...

• What is a plane?
• The plane is perpendicular to the line that is perpendicular to it (?!?!!?!)

The enemy of my enemy...

• What is a plane?
• The plane is perpendicular to the line that is perpendicular to it (?!?!!?!)

Quick review

How can you tell if two vectors $\mathbf a$ and $\mathbf b$ are perpendicular?

Quick review

How can you tell if two vectors $\mathbf a$ and $\mathbf b$ are perpendicular?

So how do you write the equation describing "the set of all endpoints of vectors $\mathbf b$ that are perpendicular to a fixed vector $\mathbf a$"?

Quick review

How can you tell if two vectors $\mathbf a$ and $\mathbf b$ are perpendicular?

So how do you write the equation describing "the set of all endpoints of vectors $\mathbf b$ that are perpendicular to a fixed vector $\mathbf a$"?

Don't look at the next slide if you don't want to see the answer!

Quick review

How can you tell if two vectors $\mathbf a$ and $\mathbf b$ are perpendicular?

So how do you write the equation describing "the set of all endpoints of vectors $\mathbf b$ that are perpendicular to a fixed vector $\mathbf a$"?

If $\mathbf a=\langle \alpha,\beta,\gamma\rangle$ then the equation is

$$\alpha x+\beta y+\gamma z = 0.$$

Quick review

How can you tell if two vectors $\mathbf a$ and $\mathbf b$ are perpendicular?

So how do you write the equation describing "the set of all endpoints of vectors $\mathbf b$ that are perpendicular to a fixed vector $\mathbf a$"?

If $\mathbf a=\langle \alpha,\beta,\gamma\rangle$ then the equation is

$$\alpha x+\beta y+\gamma z = 0.$$

Example: if $\mathbf a=\langle 1, 2, -1\rangle$, you get $x+2y-z=0$.

Quick review

How can you tell if two vectors $\mathbf a$ and $\mathbf b$ are perpendicular?

So how do you write the equation describing "the set of all endpoints of vectors $\mathbf b$ that are perpendicular to a fixed vector $\mathbf a$"?

If $\mathbf a=\langle \alpha,\beta,\gamma\rangle$ then the equation is

$$\alpha x+\beta y+\gamma z = 0.$$

Example: if $\mathbf a=\langle 1, 2, -1\rangle$, you get $x+2y-z=0$.

What shape is that?

Piglet of calculus conjectures

Conundrum: translation

• This is OK if the plane can be anchored like vectors can at $(0,0,0)$.
• If not, we have to take what we just did and translate it in space (i.e., move it away from $(0,0,0)$).
• This is just like making the plane $z=4$ by translating the $xy$-plane up $4$ units: the plane $z=4$ is not the set of endpoints of vectors perpendicular to $\mathbf k$, just a parallel translation of it.

Let's do one together

Describe the plane $x-3y+47z-28=0$ using vectors.

• Normal vector: $\langle 1,-3,47\rangle$. How did I know?
• Trick: always just use the coefficients of $x$, $y$, and $z$
• To translate: find one solution by eyeballs. A solution: $(-16,1,1)$.
• So the plane is the set of endpoints of vectors $\mathbf v$ such that

$$(\mathbf v-\langle-16,1,1\rangle)\cdot\langle 1,-3,47\rangle=0.$$

• Another way to say it: it is what you get when you take the set of vectors perpendicular to $\langle 1, -3, 47\rangle$ and translate them all by $\langle -16, 1, 1\rangle$ (and then just keep the set of endpoints)

Practice

Describe the plane $3x-4y-5z=6$ using vectors.

Who cares?

• Using this approach, you can prove that any plane is the set of solutions of a linear equation in $x,y,z$ (see book!).
• This gives us a way to get a grip on the intersection of two planes.

Example

• Describe the intersection of the planes $x-2y-z=0$ and $2x-y+z=6$.
• Perpendicular vectors: $\langle 1,-2,-1\rangle$ and $\langle 2,-1,1\rangle$
• Common solution: $(2,0,2)$
• Thus, the line of intersection is the set of vectors $\mathbf v$ such that

  $$(\mathbf v-\langle 2,0,2\rangle)\cdot\langle 1,-2,-1\rangle=0$$

  and

  $$(\mathbf v-\langle 2,0,2\rangle)\cdot\langle 2,-1,1\rangle=0.$$

• The vector $\mathbf v-\langle 2,0,2\rangle$ is perpendicular to both: cross product!
• The line is just the endpoints of vectors of the form

  $$\langle 2,0,2\rangle+t\langle 1,-2,-1\rangle\times\langle 2,-1,1\rangle,$$

  where $t$ ranges over all scalars. Parametric equations!

Last step: expand cross product

To describe $\langle 2,0,2\rangle+t\langle 1,-2,-1\rangle\times\langle 2,-1,1\rangle$, let's expand:

• $\langle 1,-2,-1\rangle\times\langle 2,-1,1\rangle=\langle -3,-3,3\rangle$
• So the line is given by the endpoints of the vectors $\{\langle 2,0,2\rangle+t\langle -3,-3,3\rangle\}$
• Parametric form: $(x,y,z)=(2-3t,-3t,2+3t)$
• As $t$ varies, this traces out the line of intersection.

More practice

• Describe the intersection of $3x+4y+5z=6$ and $y+z=0$.
• Describe the intersection of $3x+4y+5z=6$ and $6x+8y+10y=12$.
• Describe the intersection of $3x+4y+5z=6$ and $9x+12y+15z=17$.

Next time: cylinders and quadrics!