Vectors

Question 1

Two types of measurable quantity

Answer 1

Scalar

Vector

Question 2

Scalar

Answer 2

A quantity with a magnitude (size)

Question 3

Vector

Answer 3

A quantity with a magnitude and direction

Question 4

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

Answer 4

The direction remains the same, the magnitude changes

Question 5

Component form of vector

Answer 5

( x y z )

Question 6

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

Answer 6

(2, -2)

Question 7

PQ = ?

Answer 7

q - p

Question 8

How is magnitude written

Answer 8

|u| or |AB|

Question 9

How is magnitude of a vector calculated

Answer 9

Using Pythagoras

Question 10

|PQ| = ?

Answer 10

/a^2 + b^2

Question 11

What is a unit vector

Answer 11

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

Question 12

Unit vector magnitude

Answer 12

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

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

Question 13

How to find unit vector of original vector

Answer 13

Divide original vector by its magnitude

Question 14

How many unit vectors does a vector have

Answer 14

Two, +ve and -ve

Question 15

Method for unit vectors

Answer 15

Find magnitude of original vector

Divide vector by its length

Question 16

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

Answer 16

( 3 / 5, 4 / 5 )

Question 17

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

Answer 17

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

Question 18

What to use to find a component of a unit vector

Answer 18

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

Question 19

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

Find the two values of a

Answer 19

+ - /7 / 3

Question 20

Method for proving unit vector

Answer 20

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

Statement to prove

Question 21

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

Answer 21

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

Question 22

Orgin coordinates

Answer 22

0, 0

Question 23

0, 0

Answer 23

Origin

Question 24

unit vector i

Answer 24

( 1 0 0)

Question 25

( 1 0 0)

Answer 25

Unit vector i

Question 26

Unit vector j

Answer 26

( 0 1 0)

Question 27

( 0 1 0)

Answer 27

Unit vector j

Question 28

Unit vector k

Answer 28

( 0 0 1)

Question 29

( 0 0 1)

Answer 29

Unit vector k

Question 30

i, j, k =

Answer 30

i = x j = y k = z

Question 31

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

Answer 31

( 4 -8 3)

Question 32

Express 7j - 2k in component form

Answer 32

( 0 7 -2)

Question 33

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

Answer 33

2i + 5j - k

Question 34

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

Answer 34

-2i + 5k

Question 35

The vector, a, has the components (-3 0 4). Find the unit vector parallel to a in the form i, j, k.

Answer 35

(-3/5 0 4/5) = -3/5i + 4/5k

Question 36

Parallel vectors relationship

Answer 36

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

Question 37

Method for proving if 2 vectors are parallel

Answer 37

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

Question 38

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

Answer 38

3u = v

Question 39

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

Answer 39

a = 2/7b

Question 40

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

Answer 40

p = 2/-3q

Question 41

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

Answer 41

No, (4 3) ≠ ( 2 1)

Question 42

Proving collinearity of vectors method

Answer 42

1. Find vector of first 2 coordinates 2. Find vector of last 2 coordinates 3. Statement

Question 43

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

Answer 43

2BC = AB so parallel and share common point B

Question 44

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

Answer 44

Parallel since 2BC = AB and share common point B

Question 45

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

Answer 45

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

Question 46

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

Answer 46

a = 20 b = 9

Question 47

Section formula

Answer 47

na + mb/ ( m + n ) remember, A P B m n

Question 48

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

Answer 48

AP:PB = 3:2

Question 49

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

Answer 49

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

Question 50

Section formula method

Answer 50

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

Question 51

AP:PB = 2:5 Find P, given that A is (3, 4) and B is (17, 11)

Answer 51

(7, 6)

Question 52

P divides AB in the ratio 3:2 Find P given that A is (2, -3, 4) and B is (12, 7, -1)

Answer 52

P is (8, 3, 1)

Question 53

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.

Answer 53

P = (2, 1,3) Q = (2, 0, 1) R = (2, -3, -5)

Question 54

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

Answer 54

(3, -2, -5

Question 55

VABCO is a pyramid. AB = 8i + 2j + 2k AD = -2i + 10j - 2k AV = i + 7j + 7k Express CV in component form

Answer 55

( -5 -5 7)

Question 56

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

Answer 56

EK = i- 8j - 16k

Question 57

RATU, VWXY is a cuboid Express VT in terms of f, g, and h

Answer 57

- f - h + g

Question 58

Scalar product equation (3 forms)

Answer 58

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|

Question 59

3 things to remember with scalar product

Answer 59

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

Question 60

Find scalar product:

Answer 60

6

Question 61

Find scalar product

Answer 61

1/2

Question 62

Find scalar product:

Answer 62

0

Question 63

Find the scalar product

Answer 63

-1

Question 64

Find the scalar product:

Answer 64

-6

Question 65

Find the scalar product:

Answer 65

-10 root 3

Question 66

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

Answer 66

-9

Question 67

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.

Answer 67

u.v = 34

Question 68

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

Answer 68

a) (7, -1, 3) b) 54

Question 69

Angle between two vectors formula

Answer 69

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

Question 70

a = i + 2j + 2k and b = 2i + 3j + 6k. Calculate the angle between a and b (underlined a and b in all text)

Answer 70

17.8 degrees

Question 71

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

Answer 71

a.b = 0 since perp. Cos90 = 0

Question 72

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

Answer 72

p = -7

Question 73

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

Answer 73

131.8

Question 74

scalar product if 0 < theta < 90

Answer 74

+ve

Question 75

Scalar product if theta = 90

Answer 75

0

Question 76

scalar product if 90 < theta < 180

Answer 76

-ve

Question 77

a.(b + c) expanded

Answer 77

a.b + a.c

Question 78

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

Answer 78

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

Question 79

unit vector.unit vector

Answer 79

0

Question 80

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

Answer 80

a.b + a.c = 0 a is perpendicular to b + c

Question 81

Answer 81

26

Question 82

Answer 82

16

Question 83

Answer 83

-36