Chapter 10 Equations

Question 1

Unit vector

Answer 1

ā
—-
llāll

Question 2

Magnitude of a vector

Answer 2

sqrt(a^2+b^2+c^2)

Question 3

Angle between two vectors

Answer 3

cosØ=ā•b
——
llāll•llbll

Question 4

Scalar projection of b onto ā

Answer 4

comp b=ā•b
a ——
llāll

Question 5

Vector projection of b onto a

Answer 5

Proj b= ā•b
ā —— *a
llāll^2

Question 6

Vector equation of a line

Answer 6

r=ro+tv

Question 7

Parametric equation of a line

Answer 7

x=xo+at
y=yo+bt 
z=zo+ct
Through P(xo yo zo)
and vector <a></a>

Question 8

Symmetric equations of a line

Answer 8

x-xo y-yo z-zo
——=——-=——-
a b c

Question 9

Scalar equation of a plane

Answer 9

a(x-xo)+b(y-yo)+c(z-zo)=0
Through P(xo yo zo)
With normal vector ñ=<a></a>

Question 10

Vector equation of a plane

Answer 10

ñ(r-ro)=0

Question 11

Equation of a plane with just ñ vector

Answer 11

ax+by+cz+d=0

Question 12

Angle between planes

Answer 12

cosØ= ñ•ñ
————
llñll•llñll

Question 13

Length of a space curve

Answer 13

L=integral sqrt(x’(t)^2+y’(t)^2+z’(t)^2)

Question 14

Unit tangent vector of a curve

Answer 14

T(t)=r’(t)/llr’(t)ll

Question 15

Curvature K of a curve (3 equations)

Answer 15

K=lldT/dsll
K(t)=ll T’(t)/r’(t) ll
K(t)=ll r’(t)Xr’’(t)ll/ll r’(t)ll^3

Question 16

Principal unit normal vector N

Answer 16

N(t)=T’(t)/llT’(t)ll

Question 17

Binormal vector

Answer 17

B(t)=T(t)X N(t)

Question 18

Acceleration of a particle

Answer 18

ā(t)=v’(t)=r’’(t)t

Question 19

Speed of a particle

Answer 19

ll V(t) ll

Question 20

Components of acceleration

Answer 20

ā=llvll•T+K•||vll^2•N