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Higher Mathematics > Vectors > Flashcards

Vectors Flashcards

1
Q

Two types of measurable quantity

A

Scalar

Vector

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2
Q

Scalar

A

A quantity with a magnitude (size)

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3
Q

Vector

A

A quantity with a magnitude and direction

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4
Q

When multiplying a vector by a scalar, what stays the same and what changes

A

The direction remains the same, the magnitude changes

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5
Q

Component form of vector

A

( x y z )

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6
Q

given the points P (3,4), Q(5,2) and O (0,0), find the vector joining them

A

(2, -2)

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7
Q

PQ = ?

A

q - p

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8
Q

How is magnitude written

A

|u| or |AB|

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9
Q

How is magnitude of a vector calculated

A

Using Pythagoras

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10
Q

|PQ| = ?

A

/a^2 + b^2

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11
Q

What is a unit vector

A

A parallel vector to the original vector with a magnitude of 1

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12
Q

Unit vector magnitude

A

/l^2 + m^2 + n^2 = 1

l^2 + m^2 + n^2 = 1

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13
Q

How to find unit vector of original vector

A

Divide original vector by its magnitude

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14
Q

How many unit vectors does a vector have

A

Two, +ve and -ve

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15
Q

Method for unit vectors

A

Find magnitude of original vector

Divide vector by its length

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16
Q

Find the components of the unit vector, u, parallel to vector v if v = (3 4 )

A

( 3 / 5, 4 / 5 )

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17
Q

Find two unit vectors of the vector a = ( 12, 3, 4 )

A

+ - ( 12/13 , 3/13, 4/13)

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18
Q

What to use to find a component of a unit vector

A

/ l^2 + m^2 + n^2 = 1

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19
Q

u = (a, 1/3, 1/3) is a unit vector.

Find the two values of a

A

+ - /7 / 3

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20
Q

Method for proving unit vector

A

Use l^2 + m^2 + n^2 = 1

Statement to prove

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21
Q

Is (2/3 1/3 -2/3 ) a unit vector

A

Yes since l^2 + m^2 + n^2 = 1

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22
Q

Orgin coordinates

A

0, 0

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23
Q

0, 0

A

Origin

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24
Q

unit vector i

A

( 1 0 0)

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25
Q

( 1 0 0)

A

Unit vector i

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26
Q

Unit vector j

A

( 0 1 0)

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27
Q

( 0 1 0)

A

Unit vector j

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28
Q

Unit vector k

A

( 0 0 1)

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29
Q

( 0 0 1)

A

Unit vector k

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30
Q

i, j, k =

A

i = x
j = y
k = z

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31
Q

Express 4i - 8j + 3k in component form

A

( 4 -8 3)

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32
Q

Express 7j - 2k in component form

A

( 0 7 -2)

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33
Q

Express (2 5 -1) in the form of i, j, and k

A

2i + 5j - k

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34
Q

Express (-2 0 5) in the form of i, j, and k

A

-2i + 5k

35
Q

The vector, a, has the components (-3 0 4).

Find the unit vector parallel to a in the form i, j, k.

A

(-3/5 0 4/5)

= -3/5i + 4/5k

36
Q

Parallel vectors relationship

A

if vector u = ( x y ), then ku = (kx ky)

37
Q

Method for proving if 2 vectors are parallel

A
  1. Take out highest common factor in each vector
  2. Check for same vector
  3. Express vector in terms of other
38
Q

Prove u = (2 3) and v (6 9) are parallel

A

3u = v

39
Q

Show a = (6 -2) and b = (21 -7) are parallel

A

a = 2/7b

40
Q

show that p = (4 -10) and q = ( -6 15) are parallel

A

p = 2/-3q

41
Q

are c = (12 9) and d = (16 8) parallel?

A

No, (4 3) ≠ ( 2 1)

42
Q

Proving collinearity of vectors method

A
  1. Find vector of first 2 coordinates
  2. Find vector of last 2 coordinates
  3. Statement
43
Q

Prove the points A (2, 4) B (8, 6) and C (11, 7) are collinear with vectors

A

2BC = AB so parallel and share common point B

44
Q

Prove the points A (0, 1, 2) B (1, 3, -1) and C (3, 7, -7) are collinear with vectors

A

Parallel since 2BC = AB

and share common point B

45
Q

Prove the points A (2, -3, 4) B (8, 3, 1) and C (12, 7, -1) are collinear with vectors

A

AB = 2/3 BC so parallel and common point B

46
Q

The points A (-8, a, -12) B (2, -5, 3) and C (6, -15, b) are collinear. Find a and b

A

a = 20

b = 9

47
Q

Section formula

A

na + mb/ ( m + n )

remember,
A P B
m n

48
Q

P divides AB internally in the ratio 3:2. How do you write this

A

AP:PB = 3:2

49
Q

P divides AB externally in the ratio 3:2. How do you write this

A

AP:PB = 3:-2 OR -3:2

50
Q

Section formula method

A
  1. Identify ratio and points
  2. Use cross method to match up points and ratio
  3. Sub into formula
  4. Solve
51
Q

AP:PB = 2:5

Find P, given that A is (3, 4) and B is (17, 11)

A

(7, 6)

52
Q

P divides AB in the ratio 3:2

Find P given that A is (2, -3, 4) and B is (12, 7, -1)

A

P is (8, 3, 1)

53
Q

Triangle ABC has the vertices A (3, 0, 6), B (0, 3, -3) and C (1, 0, -4). P divides AB in the ratio 1:2, Q is the midpoint of AC and R divides BC externally in the ratio 2:1.

Find the coordinates of P, Q and R.

A

P = (2, 1,3)

Q = (2, 0, 1)

R = (2, -3, -5)

54
Q

Find the coordinates of the point dividing joining E (5, 2, 1) and F (9, 10, 13). It divides externally in the ratio 1:3

A

(3, -2, -5

55
Q

VABCO is a pyramid.

AB = 8i + 2j + 2k
AD = -2i + 10j - 2k
AV = i + 7j + 7k

Express CV in component form

A

( -5 -5 7)

56
Q

EABCD is a pyramid

EA = -7i - 13j - 11k
AB = 6i + 6j - 6k
AD = 8i - 4j + 4k

K divides BC in the ratio 1:3. Find EK

A

EK = i- 8j - 16k

57
Q

RATU, VWXY is a cuboid

Express VT in terms of f, g, and h

A
  • f - h + g
58
Q

Scalar product equation (3 forms)

A

a.b = |a| |b| cos0

Where 0 is the angle between a and b (underlined)

a.b = a1b1 + a2b2 + a3b3

Where a = (a1 a2 a3) and b = (b1 b2 b3)

Cos0 = a.b / |a||b|

59
Q

3 things to remember with scalar product

A

Vectors must be pointing away from the angle

The scalar product is a scalar, not a vector!

The an and b (underlined) are perpendicular, then a.b = 0

60
Q

Find scalar product:

A

6

61
Q

Find scalar product

A

1/2

62
Q

Find scalar product:

A

0

63
Q

Find the scalar product

A

-1

64
Q

Find the scalar product:

A

-6

65
Q

Find the scalar product:

A

-10 root 3

66
Q

Find scalar of a (1 2 5) and b (-1 1 -2)

A

-9

67
Q

Given P (1, 5, 8), Q (-2, 1, 3) and R (1, -6, 0), if PQ represents u and QR represents v, find u.v.

A

u.v = 34

68
Q

S is (1, 2, -3), T is (11, -3, 7)

U divides ST in the ratio 3:2. Find:

Coordinate of U

Value of SU.UT

A

a) (7, -1, 3)

b) 54

69
Q

Angle between two vectors formula

A

Cos0 = a.b / |a||b|

70
Q

a = i + 2j + 2k and b = 2i + 3j + 6k.

Calculate the angle between a and b (underlined a and b in all text)

A

17.8 degrees

71
Q

Show that an and b are perpendicular if a = (2 3 -6) and b = (-3 2 0). a and b underlined

A

a.b = 0 since perp.

Cos90 = 0

72
Q

Vectors a = 2i + 5j + k and b = pi - 2j + 3k are perpendicular. (a and b underlined)

Find the value of p

A

p = 3

73
Q

R is (4, 4, 1) S is (3, 2, 0), T is (2, 0, 1). Calculate the size of the angle RST

A

131.8

74
Q

scalar product if 0 < theta < 90

A

+ve

75
Q

Scalar product if theta = 90

A

0

76
Q

scalar product if 90 < theta < 180

A

-ve

77
Q

a.(b + c) expanded

A

a.b + a.c

78
Q

(a + b) . (c + d) expanded

A

a.c + a.d + b.c + b.d

79
Q

unit vector.unit vector

A

0

80
Q

let a = 2i - k, b = i + 2j + k, c = -j + k.

evaluate a.b + a.c

make a deduction and about a and b + c

A

a.b + a.c = 0

a is perpendicular to b + c

81
Q
A

26

82
Q
A

16

83
Q
A

-36

84
Q
A

-36

Higher Mathematics (7 decks)

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