Cross Product - Multivariable Calculus - Lecture Notes | MATH 1920, Study notes of Calculus

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The Cross Product (12.4) and Lines and Planes

in Space (12.5)

Last time in 1920...

I (^) We learned about the cross product of two vectors and the

triple scalar product and its relation to volume (and flux).

The triple scalar product (~u × ~v ) · w~ =

Last time in 1920...

I (^) We learned about the cross product of two vectors and the

triple scalar product and its relation to volume (and flux).

The triple scalar product (~u × ~v ) · w~ =

I

w 1

u 2 u 3

v 2 v 3

− w 2

u 1 u 3

v 1 v 3

• w 3

u 1 u 2

v 1 v 2

u 1 u 2 u 3

v 1 v 2 v 3

w 1 w 2 w 3

I (^) It is helpful to know some properties of determinants when

computing cross products or triple scalar products. If the

matrix B was obtained from A by interchanging two rows,

then detA = −detB. If B was obtained from A by multiplying

a row of A by a scalar, α, then detB = αdetA.

Lines and Planes

I (^) Definition A vector equation for the line L through

P 0 (x 0 , y 0 , z 0 ) parallel to ~v is

~r (t) =< x 0 , y 0 , z 0 > +t~v , −∞ < t < ∞.

Vector Equation for a Line

I (^) True or False: Once you are given a point on a line and a

direction vector, there is exactly one parametrization for the

line in space through that point and in that direction.

I (^) False. ~r 1 (t) =< 0 , 2 , 1 > +t < 1 , 1 , 1 > and

r ~ 2 (t) =< 1 , 3 , 2 > +t < − 2 , − 2 , − 2 > are two different

parametrizations for the same line in space. How can you tell?

Vector Equation for a Line

I (^) True or False: Once you are given a point on a line and a

direction vector, there is exactly one parametrization for the

line in space through that point and in that direction.

I (^) False. ~r 1 (t) =< 0 , 2 , 1 > +t < 1 , 1 , 1 > and

r ~ 2 (t) =< 1 , 3 , 2 > +t < − 2 , − 2 , − 2 > are two different

parametrizations for the same line in space. How can you tell?

I (^) The vector from (0, 2 , 1) to (1, 3 , 2) is parallel to < 1 , 1 , 1 >

and to < − 2 , − 2 , − 2 >. So these points are on each of the

lines. They must be the same line.

Parametric Equations of a Line

I (^) True or False: The equations

x = sin t, y = 3 − 2 sin t, z = 1 + sin t, −∞ < t < ∞

parametrizes the same point set as the equations

x = s, y = 3 − 2 s, z = 1 + s , −∞ < s < ∞.

Parametric Equations of a Line

I (^) True or False: The equations

x = sin t, y = 3 − 2 sin t, z = 1 + sin t, −∞ < t < ∞

parametrizes the same point set as the equations

x = s, y = 3 − 2 s, z = 1 + s , −∞ < s < ∞.

I (^) False. If we let s = sin t we see that these equations describe

some of the same points in space. However, for all

−∞ < t < ∞, − 1 ≤ sin t ≤ 1. As a result

x = sin t, y = 3 − 2 sin t, z = 1 + sin t, traces out the line

segment between (− 1 , 5 , 0) and (1, 1 , 2) over and over. In

contrast x = t

3 , y = 3 − 2 t

3 , z = 1 + t

3 , −∞ < t < ∞ does

parametrize the same line as x = s, y = 3 − 2 s, z = 1 + s ,

−∞ < s < ∞.

More Parametric Equations of a Line

I (^) How can you tell whether two lines intersect?

I (^) Answer: Use different names for the parameters (say s and t)

and see whether there are any solutions to the system

x 1 (t) = x 2 (s), y 1 (t) = y 2 (s), and z 1 (t) = z 2 (s).

More Parametric Equations of a Line

I (^) How can you tell whether two lines intersect?

I (^) Answer: Use different names for the parameters (say s and t)

and see whether there are any solutions to the system

x 1 (t) = x 2 (s), y 1 (t) = y 2 (s), and z 1 (t) = z 2 (s).

I (^) How can you tell if two lines go through the same point at the

same “time”?

Three ways to describe a line in a plane

I (^) First a very familiar way, the “equation of a line”

f (x) = y = 3x + 7

y

x

y=3x+

(0,7)

The equation of a line via “normal vectors”

I (^) By rewriting y = 3x + 7 as 3(x − 0) + (−1)(y − 7) = 0 we see

that a point (x, y ) is on the line iff

< x − 0 , y − 7 > · < 3 , − 1 >= 0.

<3,−1>

y

x

(0,7)

(x,y)

(x,y)

<x−0,y−7>

<x−0,y−7> <−3,1>

Vector equation of a line and parametric equations of a line

I (^) We can compute the direction vector and write a vector

equation of the line through the two points:

r (t) =< 1 , 10 > +t < 1 , 3 >=< 1 + t, 10 + 3t >

<1,10>

y

x

t=

t=2 <1,10>+2<1,3>

t=− (^4) <1,10>−4<1,3>

I (^) The vector equation gives us the parametric equations

x = 1 + t, and y = 10 + 3t.

Three ways to describe a plane in space

I. Using a point and a normal vector

I (^) Given a point, P(1, 2 , 3) and a normal vector, ~n =< 7 , 2 , 4 >,

a point (x, y , z) is on the plane that contains P and is

perpendicular to ~n iff < x − 1 , y − 2 , z − 3 > · < 7 , 2 , 4 >= 0,

or

7 x + 2y + 4z − 23 = 0

<7,2,4>

X Y

Z

(3,1,0) (^) (1,8,0)

(1,2,3)