Vectors

Question 1

True or false?

Two vectors in different points of space, both having the same magnitude and direction are considered the same vector

Answer 1

True

These vectors are the same as they both have same direction and magnitude

Even though they are of different points in space. They retain the same properties

Question 2

What is the arrow of a vector called?

Answer 2

The head

Question 3

Where is the tail of a vector?

Answer 3

The starting point

Question 4

True or false?

Vector addition is communitive

Answer 4

True.

Meaning order does not matter. Essentially, 2+3 and 3+2 are the same, the order does not matter.

Question 5

What is meant by vector addition being associative?

Answer 5

u+(v+w)=(u+v)+w. It doesn’t matter if you bracket it, it will give the same answer.

Question 6

How would you write “2 dimension” symbolically?

Answer 6

Always double line on the main stroke

Question 7

How would you write “3 dimension” symbolically?

Answer 7

Always double line on the main stroke

Question 8

What does PQ with an arrow above it (left to right) represent

Answer 8

A vector from point P to point Q

Question 9

What are the two ways of writing a vector in matrix form?

Question 10

How would you write the matrix of a vector in the 3rd dimension?

Answer 10

Each row represents x, y and z.

Question 11

Draw the following vectors:

u=(3, 4) , v=(-2,3) , w=(-1, -2)

Question 12

True or false?

When representing vectors in vector addition, the tail of one vector must line up with the head of another.

Answer 12

True

Question 13

Given two vectors of the form v=(v1, v2) and w=(w1, w2)

How would you add these two vectors?

Answer 13

Question 14

Given that u=(2, 3) and v=(-4, 2)

determine the vector sum and draw it.

Answer 14

Question 15

What is a zero vector?

Answer 15

It is a vector with zero magnitude and undefined direction.

It can be represented geometrically as a point

Question 16

Given t is a scaler and v is a vector. Is tv considered scaler multiplication of the vector?

Answer 16

Yes

Question 17

How would you represent tv in matrix form

(given t is a scaler and v is a vector)

Answer 17

Note that it is the same procedure in 3 dimensions

Question 18

True or false?

Vector subtraction is a combination of vector addition and scaler multiplication

Answer 18

True

v - w = v + (-w)

Question 19

Given u=(-4, 2)

Determine the following scaler multiple: 2u

Answer 19

(-8, 4)

Question 20

Given u=(2, -1) and v=(-3, -4)

Determine the resultant vector of u-v

Answer 20

u-v=(5, 3)

Question 21

Given u=(2, -1) and v=(-3, -4)

Determine the resultant vector of v-2u

Answer 21

v-2u=(-7, -2)

Question 22

What is a position vector?

Answer 22

A vector that starts at the origin. (tail at the origin)

O is the origin

Question 23

What is the norm of a vector?

Answer 23

The shortest distance between the starting point and ending point.

It is also called the magnitude and is represented as ||v||

Question 24

How would you calculate the norm of a vector in the 2nd dimension?

Answer 24

The norm/magnitude of a vector (2 dimension) is calculated as the image shows.

Where v1 is the horizontal magnitude and v2 is the vertical magnitude.

Question 25

How would you calculate the norm of a vector in the 3rd dimension?

Answer 25

The norm/magnitude of a vector (3 dimension) is calculated as the image shows.

Question 26

# True or false? ||v+w|| = ||v|| + ||w||

Answer 26

False. ## Footnote Meaning the magnitude of the resultant vector cannot be calculated by adding them

Question 27

What is a unit vector?

Answer 27

A vector with a norm/magnitude of 1

Question 28

How would you represent a unit vector?

Answer 28

A hat above the letter:

Question 29

What are the two important unit vectors in the 2nd dimension?

Answer 29

i and j

Question 30

How would you calculate the unit vector of a given vector v?

Answer 30

Question 32

What are the three important unit vectors in the 3rd dimension?

Answer 32

i, j and k

Question 33

How would you write v=(a,b, c) in component form?

Answer 33

Question 34

# Draw the following component vectors in geometric form v=2i, 3j and w=-4i+3j

Question 35

The vector `v` in R² has magnitude 4 and direction 2π/3. Write `v` in component form and in matrix form.

Answer 35

Use the formula **v=||v||cosθi + ||v||sinθj**

Question 36

# Given that u=(3,4) Determine û

Answer 36

û=(3/5, 4/5)

Question 37

# Given v=(2, -1, 4) Determine v̂

Answer 37

v̂=(2/sqrt21, -1/sqrt21, 4/sqrt21)

Question 38

Determine the matrix form of each of the vectors drawn below:

Answer 38

ṵ=(3,2) v=(-5, -2) w=(-3, 4)

Question 39

What is the scalar product also known as?

Answer 39

The dot product or inner product

Question 40

# True or false? The scalar product returns a scalar as an output

Answer 40

True. | It returns a scalar when two vectors are multiplied

Question 41

What is the formula for dot product when the magnitude of the vectors are given?

Answer 41

Question 42

What is the formula for dot product when the matrix form of the vectors are given? (in R²)

Answer 42

`v`⋅`w`=`v`_*x*`w`_*x* + `v`_*y*`w`_*y*

Question 43

# True or false? *Scalar multiplication* and *The scalar product* are the same thing

Answer 43

False. ## Footnote In *scalar multiplication*, a scalar is multiplied to a vector to **produce a vector**. While in the *scalar product*, two vectors are multiplied together to **produce a scalar**

Question 44

In the scalar product formula, what angle is used?

Answer 44

The angle between the two vectors that is no more than 180° **(0≤θ≤π)**

Question 45

The vector `ṵ` has a norm of 5 and direction 5π/4 radians. The vector `v` has a norm of 3 and direction 7π/12 radians. Calculate `ṵ`⋅`v`

Answer 45

-15/2

Question 46

# The vector `v` = (-3,2) lies in the 2nd quadrant. Find a vector in the 3rd quadrant that is perpendicular to `v`

Answer 46

(-2, -3) ## Footnote Since they are perpendicular, u⋅v=0. Then once we get to the -3u₁ + 2u₂=0 part, we just test values (trial and error) that equal to 0. Or alternatively, you could use gradients, finding the original gradient then finding the gradient to the normal

Question 47

# True or false? Dot product is commutative

Answer 47

True, meaning the order of how the vectors are written does not matter

Question 48

# In scalar product, what expression is produced by the following? `v`⋅`v`

Answer 48

||`v`||²

Question 49

What does orthogonal mean?

Answer 49

When something is right angled *(or even perpendicular)*

Question 50

# In scalar multiplication If the angle is between 0 and 90°, is u⋅v below or above 0?

Answer 50

Above 0

Question 51

# In scalar multiplication If the angle is between 90° and 180°, is u⋅v below or above 0?

Answer 51

Below 0

Question 52

# In scalar multiplication If the angle is 90°, is u⋅v below or above 0?

Answer 52

It is 0

Question 53

Calculate the angle (*in both radians and degrees*) between vectors `u`=(4,3,1) and `v`=(2,-3,-2)

Answer 53

Question 54

# True or false? For vectors v, u and w: `(u + v) ⋅ w = u⋅w + v⋅w`

Answer 54

True | Meaning that scalar product has a distrubutive property

Question 55

# True or false? For vectors v and w and any scalar t: `(tv) ⋅ w = t(v⋅w)`

Answer 55

True

Question 56

What is the formula used in vector product?

Answer 56

Question 57

# True or false? Given vectors v, w and the vector product v⋅w=u u must be perpendicular to v and w

Answer 57

True

Question 58

What is another name for vector product?

Answer 58

Cross product

Question 59

# In vector product How would you find the direction of the resultant vector?

Answer 59

Use right hand rule. 1. Put your hand along the first vector in the formula v⋅w 2. Point all your fingers towards the direction of that vector 3. Then, point your thumb in a way where your right hand can curl towards the second vector 4. The direction your thumb points in the end is the direction of the resultant vector ## Footnote You could also use the screw rule

Question 60

Just screw rule revision (not a question)

Answer 60

Question 61

# Consider the unit vectors i, j and k in 3-space Compute the nine vector products to fill the following table | In vector product

Answer 61

You can use the i j k circle. *Basically put i, j, k in a circular patern and order them counter clockwise. An example, i⋅j: You start from i then move to j counter clockwise and you are only left with k. Note that if you move clockwise, it will become negative as indicated by the red drawing*

Question 62

# True or false? For non-zero vectors, "||`u` ⋅ `v`|| = 0 if `u` and `v` are parallel or antiparallel". | (In vector product)

Answer 62

False. The statement is for non-zero vectors, "||`u` ⋅ `v`|| = 0 _if and only if_ `u` and `v` are parallel or antiparallel."

Question 63

# True or false? In vector product, `u` ⋅ `v` = -`v` ⋅ `u`

Answer 63

True ## Footnote Meaning reversing sign of the second vector while reversing the order will produce the same vector as the original calculation

Question 64

# True or false? Vector product is commutative

Answer 64

False. Meaning the order **does** matter. ## Footnote `v` ⋅ `u` is not equivalent to `u` ⋅ `v`

Question 65

Is vector product associative or not?

Answer 65

It is not associative meaning where you bracket will matter

Question 66

# True or false? Given vectors u and v and any scalar, t: t(u⋅v) = (tu) ⋅ v = u ⋅ (tv) | In vector product

Answer 66

True. | Also note the order

Question 67

# True or false? In vector product, u ⋅ (v+w) = u ⋅ v + u ⋅ w

Answer 67

True but also note that when it is (u + v) ⋅ w = u ⋅ w + v ⋅ w | Just remember that order matters

Question 68

Answer 68

Question 69

Prove that for non-zero vectors,||`u` ⋅ `v`|| = 0 _if and only if_ `u` and `v` are parallel or antiparallel.

Answer 69

Fix this