Vectors

Question 1

“Moved five meters”

Vector or Scalar?

Answer 1

Scalar

Only a
magnitude/size.

A scalar is a
magnitude/size
alone.

Question 2

“Moved five meters to the right”

Vector or Scalar?

Answer 2

Vector

Both a
magnitude/size,
and a
direction.

A vector requires a
magnitude/size
and a
direction.

Question 3

“Moved a distance of . . .”

Vector or Scalar?

Answer 3

Scalar,
probably.

Distance alone
tends to be a scalar quantity.

Question 4

“Displaced by . . .”

Vector or Scalar?

Answer 4

Vector.

“Displaced”
is a vector term.

Often used like “Displaced five meters to the right.”

Question 5

“Speed”

Vector or Scalar?

Answer 5

Scalar,
probably.

Speed tells you
distance/time, so without a
direction,
it’s a scalar quantity.

Question 6

“Velocity”

Vector or Scalar?

Answer 6

Vector.

“Velocity” implies a
direction and a
speed,
so it’s a
vector quantity.

Question 7

The number 5.

Can this represent a
vector?

Answer 7

No.

It could be a magnitude, but not a direction.

Question 8

The angle measure 5°.

Can this represent a
vector?

Answer 8

No.

It could be a direction, but not a magnitude.

Question 9

The point (5, 5).

Can this represent a
vector?

Answer 9

Yes.

If it’s relative to the origin,
then you have a
magnitude and a
direction.

Question 10

A vector is written as
(a, b).

What
form
is that?

Answer 10

Component form

The vector is treated as a
point on the coordinate plane, or as a
directed line segment on that plane.

The components are the vector’s
x- and y-coordinates.

Question 11

A vector is written as
aî + bĵ.

What
form
is that?

Answer 11

Unit vector form

With
vector addition and
scalar multiplication,
any two-dimensional vector can be represented as a
combination of the unit vectors.

Example:

(3, 4) = 3î + 4ĵ

Question 12

A vector is written as
_→
|| u ||, Θ.

What
form
is that?

Answer 12

Magnitude and direction form

Magnitude:
the
length of the line segment

Direction:
the
angle the line forms with the
positive x-axis

Question 13

How do you know whether

• *vectors** are
• *equivalent**?

Answer 13

They have the

• *same** magnitude and
• *direction**.

Vectors are defined as a magnitude and a direction, so these two attributes must be the same if the vectors are the same.

Question 14

If vectors have the

same magnitude and direction,

then they are_____.

Answer 14

Equivalent

Question 15

If

_→
AB = (−5, 4)

then

−5 is the _____
of the vector.

Answer 15

x-component

Question 16

If

_→
AB = (−__5, 4)

then

4 is the _____
of the vector.

Answer 16

y-component

Question 17

If

A = (7, 2)
and
B = (17, −3),

then

_→
AB = (__ , __)

Answer 17

(10, −5)

_→
v = (Δx, Δy)

= (x_f − x_i, y_f − y_i)

_→
AB = (17 − 7, −3 − 2)

= (10, −5)

Question 18

If

A = (7, 2)
and
B = (17, −3),

then

_→
BA = (__ , __)

Answer 18

(−10, 5)

_→
v = (Δx, Δy)

= (x_f − x_i, y_f − y_i)

_→
BA = (7 − 17, 2 − (−3))

= (−10, 5)

Question 19

What does

_→
|| u ||

mean?

Answer 19

The magnitude of
_→
u

Question 20

If

_→
AB = (−1, 4),

then

_→
|| AB || = _____.

Answer 20

√(17)

_→
|| v || = √((Δx)², (Δy)²)
(the Pythagorean theorem)

_{​ →}
AB = (−1, 4)

_→
|| AB || = √((Δx)² + (Δy)²)

= √((−1)² + (4)²)

= √(1 + 16)

= √(17)

Question 21

If

_→
w = (2, −1),

then

_→
−2w = _____?

Answer 21

(−4, 2)

To
multiply a vector by a scalar,
you
multiply each component by the scalar.

Example:

_→
w = (2, −1)

_→
−2w = (−2 • 2, −2 • −1)

= (−4, 2)

Question 22

If

_{→ →}
v = (x, y), || v || = 4,
and
_→
w = (−2x, −2y),

then

_→
|| w || = _____?

Answer 22

8

Magnitude is always
positive.

Question 23

_→→
a = (−5, 3) and b = (−1, −2),

so

_→→
a + b = _____?

Answer 23

(−6, 1)

To add vectors, you add
x-components to x-components and
y-components to y-components.

Question 24

_→→
a = (−5, 3) and b = (−1, −2),

so

_→→
a − b = _____?

Answer 24

(−4, 5)

To subtract vectors, you subtract
x-components from x-components and
y-components from y-components.

Question 25

_→→ _a = (4, −1)_ and _b = (1, 2)_. **Visualize** _→→ **a + b = \_\_\_\_\_.**

Answer 25

*Visualize the start of* *_→→ b on the end of a.*

Question 26

_{→ →} _a = (4, −1)_ and _b = (1, 2)_. **Visualize** _{→ →} **a − b = \_\_\_\_\_.**

Answer 26

*_→→→→ a − b = a + (−1)b.* *Visualize the start of _→→ −b on the end of a.*

Question 27

_→→ u = (−1, −7) and w = (3, 1) What are the **steps** to solving **_→→ 2u + (−3)w**?

Answer 27

1. **Multiply the vectors by the scalars** 2. **Add respective *x*-components and respective *y*-components**

Question 28

Graphically, when **adding or subtracting vectors**, why can you **visualize** them **head-to-tail**?

Answer 28

Because vectors are **equivalent** if their **magnitude and direction** are equivalent. The vectors can be **drawn** **anywhere**.

Question 29

_{→ →} _a + b_ is graphed below. **Visualize** the graph of _{→ →} **b + a**.

Answer 29

*Addition is commutative, so the order doesn't matter. You'll end up with the same vector.*

Question 30

_{→ →} a = [6] and b = [−4] [−2] [4] so **_→→ a + b = \_\_\_\_\_**?

Answer 30

**[2] [2]** ## Footnote _→→ a + b = [6 + −4] = [2] [−2 + 4] = [2]

Question 31

``` # _define_: unit vector ```

Answer 31

**A vector with a magnitude of one**

Question 32

What are the **unit vector components**?

Answer 32

**î and ĵ**

Question 33

What are the **unit vectors** in their **component form**?

Answer 33

**î = (1, 0) = [1] [0]** **ĵ = (0, 1) = [0] [1]**

Question 34

Given vector w, how do you **indicate** **unit vector w**?

Answer 34

**ŵ**

Question 35

If **_→ w = (4, 3)** and **magnitude of 5**, then **ŵ = \_\_\_\_\_**?

Answer 35

**(4/5, 3/5)** *Scaling each component by the _same number_ (here, the _magnitude_) leaves the _same direction_.* _{→ →} û = ( a / || u || , b / || u || ) ŵ = (4/5, 3/5)

Question 36

If _→ w = [2] [3], then what is _→ w in **unit vector form**?

Answer 36

**_→ w = 2î + 3ĵ** ## Footnote î = [1] [0] ĵ = [0] [1] [2] = 2 • [1] + 3 • [0] [3] [0] [1] = 2î + 3ĵ

Question 37

What _unit vector_ lies in the direction of **(−2, 1)**?

Answer 37

**(−2 / √(5), 1 / √(5))** *_→ If u = (a, b), then _{→ →} û = (a / || u ||, b / || u ||).* _Example_: w = (−2, 1) || w || = √( (−2)² + (1)² ) = √( (4 + 1 ) = √(5) ŵ = (−2 / √(5), 1 / √(5))

Question 38

How do you determine the **magnitude** of **(−√3, −1)**?

Answer 38

**|| (a, b) || = √(a² + b²)​** * || (a, b) || = √(a² + b²) *(Pythagorean theorem)* * || (−√3, −1) || = √((−√3)² + (−1)²) * = √(3 + 1) * = 2

Question 39

How do you determine the **direction** of **(−√3, −1)**?

Answer 39

**Θ = tan⁻¹(b/a)​** * Θ = tan⁻¹(b/a)​ * Θ = tan⁻¹(−1 / −√3)​ * =(?) 30° * NOTE: (−√3, −1) lies in Quadrant III, so must add 180° to get angle in Quadrant III * = 30° + 180° * = 210°

Question 40

How do you determine the **components** given a **magnitude of 2** and **angle of 210°**?

Answer 40

**_→→ ( || u || cosΘ , || u || sinΘ)** * _{→ →} ( || u || cosΘ , || u || sinΘ ) * = ( 2 cos(210°) , 2 sin(210°)​ ) * = ( 2 (−√(3)/2), 2 (−1/2)) * = (−√3, −1)