MAT 1236Calculus III

Section 12.5 Part I

Equations of Line and Planes

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HW…

WebAssign 12.5 Part I

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Equations of Lines• Vector Equations

• Parametric Equations

• Symmetric Equations

Equations of Planes

Recall: Position Vectors

Given any point , is the position vector of P.

To serve as a position vector, the initial point O of the vector is fixed.

1 2,OP a a�������������� 1 2,P a a

Equations of Lines

In 2D, what kind of info is required to determine a line?• Type 1:

• Type 2:

Q: How to extend these ideas?

Vector Equations

Ingredients• A (fixed) point on the line

• A (fixed) vector v=<a,b,c> parallel to the line

Any vector parallel to the line can be represented by ________________

The position vector of a (general) point on the line can be represented by ________________

0 0 0 0, ,P x y z

, ,P x y z

Parametric Equations

0

0 0 0, , , , , ,

r r tv

x y z x y z t a b c

, ,v a b c

Example 1

Find a vector equation and parametric equations for the line that passes through the point (1,1,5) and is parallel to the vector <1,2,1>.

0

Vector Equation

r r tv

Example 1

0

Vector Equation

r r tv

Example 1: Parametric Equation

Can you recover (1,1,5) and <1,2,1> from the parametric equation?

1 , 1 2 , 5x t y t z t

Remarks

As usual, parametric equations are not unique (e.g. v1=<-2,-4,-2> gives another parametric equation.)

Example 1: Symmetric Equation

1 , 1 2 , 5x t y t z t

Example 1: Symmetric Equation

Can you recover (1,1,5) and <1,2,1> from the symmetric equation?

1 , 1 2 , 5x t y t z t

What if…

1 , 1 2 , 5x t y t z t

If one of the component is a constant, then…

3 Possible Scenarios

Given 2 lines in 3D, they are either•

•

•

Example 2

Show that the 2 lines are parallel.

1

2

: 1 , 1 2 , 5

: 5 2 , 3 4 , 2

L x t y t z t

L x s y s z s

Example 3

Find the intersection point of the 2 lines

(The lines intersect if there is a pair of parameters (s,t) that gives the same point on the two lines.)

1

2

: 2 , 3 4 , 1

: 1 , 3 ,

L x t y t z t

L x s y s z s

1, 0

0,3,1

s t

Expectations

You are expected to carefully explain your solutions. Answers alone are not sufficient for quizzes or exams.

Example 4

Show that the two lines are skew.

1

2

: 1 , 2 3 , 4

: 2 , 3 , 3 4

L x t y t z t

L x s y s z s

Example 4

Show that the two lines are skew.

1

2

: 1 , 2 3 , 4

: 2 , 3 , 3 4

L x t y t z t

L x s y s z s

1. Show that the two lines are not parallel.

2. Show that the two have no intersection points.

13 6 5

11 8(1), (2) ,

5 5

i j k

t s

Expectations

To show that two lines are non-parallel, you are expected to show that the cross product of the two (direction) vectors is a non-zero vector.

Do not substitute s and t directly into the 3rd equation. You are expected to compute the values of the two sides separately and compare the values.