1. Revision of Vectors

Question 1

Vector

Answer 1

-magnitude and direction

Question 2

Scalar

Answer 2

-magnitude only

Question 3

Vector Rules

Addition

Answer 3

|a + |b = |b + |a

|a + (-|a) = 0

Question 4

Vector Rules

Components

Answer 4

|a = a1e1 + a2e2 + a3e3 = a1i^ + a2j^ + a3k^ = (a1, a2, a3)

Question 5

When are vectors equal?

Answer 5

vectors are equal if:

• they have the same magnitude and direction
• all the components are the same

Question 6

True Vectors

Answer 6

• true vectors are coordinate-invariant
• their properties do not depend on the choice of coordinate basis with which to represent them
  e. g. y is not since its value depends on the sirection of e2

Question 7

Magnitude of a Vector

Answer 7

|a| = √(a1² + a2² + a3²)

Question 8

Unit Vetor

Answer 8

vector with magnitude of 1

Question 9

Direction of a Vector

Answer 9

the direction of a vector is the unit vector in the same direction

a^ = |a / |a|

Question 10

Scalar Product

Answer 10

|a . |b = |a||b| cosθ

Question 11

Vector Product

Answer 11

|a x |b = |a||b| sinθ

Question 12

Equations of Lines

Answer 12

|r = |a +λ|u Or |r x |u = |a x |u

equation of a line in through |a and in the direction of |u

Question 13

Equations of Planes

Answer 13

|r = |a + |b
where |a is the vector from the origin to point A on the plane and |b is a vector along the plane from point A to P on the plane
|r.|n = |r.(|a+|b) = |r.|a + |r.|b
but |r and |b are perpendicular so their dot product is zero
|r . |n = |a. |n

Question 14

Scalar Triple Product

Answer 14

|a . |b x |c = |a . (|b x |c)

• produces a scalar from three vectors
• you must do the vector product first

Question 15

Vector Triple Product

Answer 15

ax(bxc) doesnt equal (axb)xc

ax(bxc) = (a.c)b - (a.b)c
(axb)xc = (a.c)b - (b.c)a

Question 16

Minimum Distance From Origin to Point P on a Line

Answer 16

-occurs when |r is perpendicular to the direction of the line |u
| |r x |u | = | |a x |u | = d | |u |
so
d = | |a x |u |/||u|

Question 17

Minimum Distance From Origin to Point P on a Plane

Answer 17

-the minimum distance occurs when |r is parallel to |n, in this case:
|r.|n = |a.|n = d||n|
so:
d = ||a.|n|/||n|

Question 18

Properties of the Scalar Triple Product

Answer 18

a. bxc = axb.c
a. bxc = b.cxa = c.axb (cyclic permutations)
a. bxc = 0 if a, b, c, are coplanar
a. bxc is the volume of the parallelepipe with edges a, b, c
a. bxc is sometimes written [a,b,c]