PHYSICS - CLUTCH NON-CALC CH 03:...

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PHYSICS - CLUTCH NON-CALC

CH 03: VECTORS

CONCEPT: REVIEW OF VECTORS AND SCALARS

● Remember: When you take measurements, you always get the size (Magnitude, how much).

- Measurements with direction → [ Vectors | Scalars ]; without direction → [ Vectors | Scalars ]

Measurement Quantity Magnitude? Direction? Vector/Scalar

“It’s 60°F outside” Temperature [ Vector | Scalar ]

“I pushed with 100N north” Force [ Vector | Scalar ]

“I walked for 10 m” Distance [ Vector | Scalar ]

“I walked 10 m east” Displacement [ Vector | Scalar ]

“I drove at 80 mph” Speed [ Vector | Scalar ]

“I drove 80mph west” Velocity [ Vector | Scalar ]


CH 03: VECTORS

CONCEPT: INTRO TO VECTOR MATH

● Adding/subtracting scalars is easy. But vectors have direction, so math with vectors is sometimes not as straightforward.

- Because vectors have direction, they’re drawn as ________________.

EXAMPLE: For each of the following situations, draw your displacement vectors and calculate the total displacement:

COMBINING

SCALARS

3 kg 4 kg

“You combine a 3kg & 4kg box”

COMBINING PERPENDICULAR

VECTORS

Total Mass: 3kg + 4kg = _______ Total Displacement: __________

“You walk 3m right, then 4m up”

● Forms _____________.

- just TRIANGLE MATH

● Simple Addition

(a) You walk 10m to the right, and then 6m to the left (b) You walk 6m to the right, and then 8m down

COMBINING PARALLEL VECTORS

“You walk 3m right, then 4m right”

Total Displacement: __________

● Add just like normal numbers


CH 03: VECTORS

PRACTICE: Two perpendicular forces act on a box, one pushing to the right and one pushing up. An instrument tells you

the magnitude of the total force is 13N. You measure the force pushing to the right is 12N. Calculate the force pushing up.


CH 03: VECTORS

CONCEPT: ADDING VECTORS GRAPHICALLY

● Vectors are drawn as arrows and are added by ______________ the arrows (tip-to-tail).

● The RESULTANT vector (�⃗⃗� or �⃗⃗� ) is always the SHORTEST PATH from the start of the first vector → end of the last.

- Adding vectors does NOT depend on the order (commutative), so �⃗⃗� + �⃗⃗� = �⃗⃗� + �⃗⃗� .

EXAMPLE: Find the magnitude of the Resultant Vector �⃗⃗� = �⃗⃗� + �⃗⃗� .

⇔

ADDING PERPENDICULAR VECTORS

3m

4m

ADDING ANY VECTORS

�⃗⃗�

�⃗⃗�

�⃗⃗� + �⃗⃗� �⃗⃗� + �⃗⃗�

𝒚

𝒙

�⃗⃗�

�⃗⃗�

Resultant Vector: (Total Displacement)

____________


____________


____________


CH 03: VECTORS

PRACTICE: A delivery truck travels 8 miles in the +x-direction, 5 miles in the +y-direction, and 4 miles again in the

+x-direction. What is the magnitude (in miles) of its final displacement from the origin?

𝒚

𝒙


CH 03: VECTORS

EXAMPLE: Find the magnitude of the Resultant Vector �⃗⃗� = �⃗⃗� + �⃗⃗� + �⃗⃗� .

𝒚

𝒙 −𝒙

−𝒚

�⃗⃗�

�⃗⃗�

�⃗⃗�


CH 03: VECTORS

CONCEPT: SUBTRACTING VECTORS GRAPHICALLY

● Subtracting vectors is exactly like adding vectors tip-to-tail, but one (or more) of the vectors gets _______________.

EXAMPLE: Find the magnitude of the Resultant Vector �⃗⃗� = �⃗⃗� − �⃗⃗� .

⇔

ADDING VECTORS

�⃗⃗� − �⃗⃗� �⃗⃗� − �⃗⃗�

SUBTRACTING VECTORS

�⃗⃗� + �⃗⃗� �⃗⃗� + �⃗⃗�

● When adding, order [ DOES | DOES NOT ] matter

● “Negative” vector: SAME magnitude, ____________ direction

● When subtracting, order [ DOES | DOES NOT ] matter

𝒚

𝒙

𝒚

𝒙

𝒚

𝒙

�⃗⃗�

�⃗⃗�

𝒚

𝒙

Resultant → shortest path: (Total Displacement)

_______________


_______________


_______________

�⃗⃗�

�⃗⃗�

𝒚

𝒙

�⃗⃗�

�⃗⃗�


CH 03: VECTORS

PRACTICE: Find the magnitude of the Resultant Vector �⃗⃗� = �⃗⃗� − �⃗⃗� − �⃗⃗� .

𝒚

𝒙 −𝒙

−𝒚

�⃗⃗� �⃗⃗�

�⃗⃗�


CH 03: VECTORS

CONCEPT: ADDING MULTIPLES OF VECTORS

● When you multiply a vector by a number (𝐴 → 2𝐴 ), the magnitude (length) changes but NOT the direction.

EXAMPLE: Find the magnitude of the Resultant Vector �⃗⃗� = 𝟑�⃗⃗� − 𝟐�⃗⃗� .

𝒚

𝒙

ADDING VECTORS

ADDING MULTIPLES OF VECTORS

�⃗⃗�

�⃗⃗�

● Multiplying by > 1 [ increases | decreases ] magnitude/length

● Multiplying by < 1 [ increases | decreases ] magnitude/length

�⃗⃗� + �⃗⃗� 𝟐�⃗⃗� + 𝟎. 𝟓�⃗⃗� 𝒚

𝒙

Resultant Vector → Shortest Path: (Total Displacement)

_______________

𝒚

𝒙

�⃗⃗� �⃗⃗�

Resultant Vector → Shortest Path: (Total Displacement)

_______________


CH 03: VECTORS

CONCEPT: VECTOR COMPOSITION AND DECOMPOSITION

● You’ll need to do vector math without using grids/ squares.

- Vectors have magnitude (length), direction (angle 𝜽𝒙), and components (legs).

EXAMPLE: For each of the following, draw the vector and solve for the missing variable(s).

VECTOR DECOMPOSITION

𝑨𝒙 = __________

𝑨𝒚 = __________

VECTOR COMPOSITION

𝑨 = √𝑨𝒙 𝟐 + 𝑨𝒚

𝟐

𝜽𝒙 = ____________

VECTOR COMPOSITION VECTOR DECOMPOSITION

● Use SOH-CAH-TOA to decompose �⃗⃗� →components 𝐴𝑥 & 𝐴𝑦.

- Angle 𝜽𝒙 must be drawn to nearest ________

1D Components → 2D Vector (Magnitude & Direction)

● Components 𝑨𝒙 & 𝑨𝒚 combine → magnitude �⃗⃗�

- Points in direction 𝜽𝒙

a) Ax = 8m, Ay = 6m, 𝑨 = ? θx = ? b) B = 13m, θx = 67.4°, Bx = ? By = ?

+𝒚

+𝒙 3

4

2D Vector (Magnitude & Direction) → 1D Components

θx=53°

5

+𝒚

+𝒙

𝒚

𝒙

𝒚

𝒙


CH 03: VECTORS

EXAMPLE: A vector A has y-component of 12 m makes an angle of 67.4° with the positive x-axis. (a) Find the magnitude of

A. (b) Find the x-component of the vector.

Vector Composition

(Components→Vector)

Vector Decomposition

(Vector→Components)

𝑨 = √𝑨𝒙 𝟐 + 𝑨𝒚

𝟐

𝜽𝑿 = 𝐭𝐚𝐧−𝟏 (𝑨𝒚

𝑨𝒙)

𝑨𝒙 = 𝑨 𝒄𝒐𝒔(𝜽𝑿)

𝑨𝒚 = 𝑨 𝒔𝒊𝒏(𝜽𝑿)


CH 03: VECTORS

CONCEPT: VECTOR ADDITION BY COMPONENTS

● You’ll need to add vectors together and calculate the magnitude & direction of the resultant without counting squares.

EXAMPLE: You walk 5m at 53° above the +x-axis, then 8m at 30° above the +x-axis. Calculate the magnitude & direction

of your total displacement.

VECTOR ADDITION

1) Draw & connect vectors tip-to-tail

2) Draw Resultant & components

3) Calculate ALL X&Y components

4) Combine X & Y components according to R equation

5) Calculate R and 𝜃𝑅

Vector Composition




𝑹 = √𝑹𝒙 𝟐 + 𝑹𝒚

𝟐

𝜽𝑿 = 𝐭𝐚𝐧−𝟏 (𝑹𝒚

𝑹𝒙)


𝑨𝒚 = 𝑨 𝒔𝒊𝒏(𝜽𝑿) x y

�⃗⃗�

�⃗⃗�

�⃗⃗� = ______

ADDING VECTORS GRAPHICALLY (WITH SQUARES)

ADDING VECTORS BY COMPONENTS (WITHOUT SQUARES)

+𝒚

+𝒙

+𝒚

+𝒙

�⃗⃗�

�⃗⃗�


CH 03: VECTORS

EXAMPLE: Vector �⃗⃗� has a magnitude of 10m at a direction 40° above the +x-axis. �⃗⃗� has magnitude 3 at a direction 20°

above the x-axis. Calculate the magnitude and direction of �⃗⃗� = �⃗⃗� − 𝟐�⃗⃗� .

VECTOR ADDITION

1) Draw & connect vectors tip-to-tail

2) Draw Resultant & components

3) Calculate ALL X&Y components

4) Combine X & Y components according to R equation

5) Calculate R and 𝜃𝑅

Vector Composition




𝑹 = √𝑹𝒙 𝟐 + 𝑹𝒚

𝟐

𝜽𝑿 = 𝐭𝐚𝐧−𝟏 (𝑹𝒚

𝑹𝒙)


𝑨𝒚 = 𝑨 𝒔𝒊𝒏(𝜽𝑿)

𝒚

𝒙


CH 03: VECTORS

CONCEPT: DOING MATH WITH VECTORS IN ANY QUADRANT (MORE TRIG)

● You’ll need to do math with vectors in ALL Quadrants, not just Quadrant 1.

EXAMPLE: Calculate a) the components and b) the absolute angle for the given vector �⃗⃗� :

Signs of Magnitudes & Components of Vectors:

● Magnitudes → Always positive, but Components may be + or − - Positive Components = pointing [ UP | DOWN ] or [ RIGHT | LEFT ]

- Negative Components = pointing [ UP | DOWN ] or [ RIGHT | LEFT ]

When given a Non-Reference Angle:

● Remember: We always use the Reference Angle 𝜽𝒙 to calculate components:

Ax = A cos(𝜽𝒙) Ay = A sin(𝜽𝒙)

- All right angles add up to 90°, so we’ll use this simple equation to get 𝜽𝒙:

Calculating the Absolute Angle (Positive Angle from +x axis) from the Arctangent

● Taking arctan of components [𝜃𝑥 = tan−1 (|𝐴𝑦|

|𝐴𝑥|)] always gives reference angle 𝜽𝒙.

- Remember to always plug in positive value of components!

- To find the Absolute Angle, work your way back to +x-axis (0°)

+𝒚

+𝒙

−𝒚

−𝒙

Quadrant 1 Quadrant 2

Quadrant 4 Quadrant 3

�⃗⃗� =5 �⃗⃗� =5

�⃗⃗� =5 �⃗⃗� =5

𝑨𝒙= 4

𝑨𝒚= 3

𝑩𝒙= 4 𝑩𝒚= 3

𝑫𝒙= 3

𝑫𝒚= 4 𝑪𝒙= 3

𝑪𝒚= 4

+𝒚

+𝒙

−𝒚

−𝒙

10°

�⃗⃗�

________________

+𝒚

+𝒙

−𝒚

−𝒙

�⃗⃗� =5

𝑨𝒙= −4

𝑨𝒚= +3

+𝒚

+𝒙

−𝒚

−𝒙

�⃗⃗� =13

22.6°


CH 03: VECTORS

PRACTICE: FINDING VECTOR COMPONENTS

Vector F is 65 m long, directed 30.5° below the positive x-axis. (a) Find the x-component, Fx. (b) Find the y-component, Fy.


CH 03: VECTORS

PRACTICE: VECTOR COMPOSITION IN ALL QUADRANTS

The vector A represented is by the pair of components Ax = -77 cm , Ay = 36 cm. (a) Find the magnitude of vector A. (b)

Find the absolute angle of this vector.


CH 03: VECTORS

CONCEPT: DESCRIBING DIRECTIONS VECTORS WITH WORDS (MORE TRIG)

● Many problems will use different words to describe the directions of vectors:

1) Counterclockwise angles are [ + / − ]; Clockwise angles are [ + / − ]

- However, reference angle 𝜽𝒙 for component equations is always a positive angle relative to nearest x-axis.

2) Angles North/South/West/East (e.g. 30° north of east): Draw arrow in 2nd direction, curve towards 1st

EXAMPLE: Draw each vector and calculate the x-component

𝑵

𝑬

𝑺

𝑾

�⃗⃗�

�⃗⃗�

+𝒙

𝑵

𝑬

𝑺

𝑾

EXAMPLE: Draw each vector and calculate its components.

a) �⃗⃗� = 5m @ +37° from -x axis b) �⃗⃗� = 5m 53° CW from +y axis

a) �⃗⃗� = 6 @ 30° North of East b) �⃗⃗� = 10 @ 53° West of South


CH 03: VECTORS

PRACTICE: HELICOPTER TRIP

A small helicopter travels 225 m across a city in a direction 53.1° south of east. What are the components of the helicopter’s trip?


CH 03: VECTORS

CONCEPT: UNIT VECTORS

● Vectors are sometimes represented using a special notation called Unit Vectors.

● Unit vectors make vector addition very straightforward:

EXAMPLE: Vector �⃗⃗� = 4𝑖̂ + 2𝑗 ̂and �⃗⃗� = −�̂� + 𝟐𝒋.̂ Draw the vectors and calculate �⃗⃗� = �⃗⃗� + �⃗⃗� in unit vector form.

Vector Addition w/ Unit Vectors

�⃗⃗� = 𝐴𝑥 �̂� + 𝐴𝑦𝒋̂ = ______________

�⃗⃗� = 𝐵𝑥 �̂� + 𝐵𝑦𝒋̂ = ______________

�⃗⃗� = �⃗⃗� + �⃗⃗� = ____________________

�̂� points in ____ direction.

𝒋̂ points in ____ direction.

𝒌 points in ____ direction.

𝒚

𝒙

Graphical Magnitude & Direction Unit Vector

𝒚

𝒙

“5m @ 53°” +𝒚

+𝒙

+𝒛

● special kind of vector that __________ in a direction

- has magnitude/length ____.

“ 3�̂� + 4𝒋̂”


CH 03: VECTORS

PRACTICE: �⃗⃗� = (4.0 m)�̂� + (3.0 m)𝒋̂ and �⃗⃗� = (−13.0 m)�̂� + (7.0 m)𝒋̂. You add them together to produce another vector �⃗⃗� .

(a) Express this new vector �⃗⃗� in unit-vector notation. (b) What are the magnitude and direction of �⃗⃗� ?


CH 03: VECTORS

EXAMPLE: Consider the three displacement vectors �⃗⃗� = (3 �̂� − 3 𝒋̂) m, �⃗⃗� = (�̂�̂ − 4 𝒋̂̂) m, and �⃗⃗� = (−𝟐 �̂� + 𝟓 𝒋̂) m.

(a) Find the magnitude and direction of D = A + B + C.

(b) Find the magnitude and direction of E = −A − B + C.


CH 03: VECTORS

CONCEPT: INTRO TO DOT PRODUCT (SCALAR PRODUCT)

● Multiplying Vectors by Scalars is simple. You’ll need to know 2 different ways to multiply Vectors by other Vectors:

1) Dot Product (Scalar Product): _______

2) Cross Product (Vector Product): _______ (covered later)

EXAMPLE: Calculate the Dot Product of �⃗⃗� and �⃗⃗� in each of the following:

+𝒚

+𝒙 −𝒙

a)

�⃗⃗� = 4 @ 0°

�⃗⃗� = 3 @ 0°

Multiples of Vectors Dot Product

4 times 3 =

4 ● =

3

Vector * Scalar (#) = Vector (number + direction) Vector • Vector = Scalar (number only, no direction)

�⃗⃗� ● �⃗⃗� = _____________ - 𝜽 = smallest angle from �⃗⃗� to �⃗⃗�

- Put calculator in degrees mode!

+𝒚

+𝒙 −𝒙

b)

+𝒚

+𝒙 −𝒙

c)

+𝒚

+𝒙 −𝒙

d)

+𝒚

+𝒙 −𝒙

e)

�⃗⃗� = 4 @ 60°

�⃗⃗� = 3 @ 0°

�⃗⃗� = 4 �⃗⃗� = 3

�⃗⃗� = 4

�⃗⃗� = 3

�⃗⃗� = 4

�⃗⃗� = 3

- Dot Product = multiplication of ____________ components.

- Negative Dot Product = components in ____________ directions.

60°

- ZERO Dot Product = components in ______________ directions.

- Always line up vectors end-to-end (tail-to-tail)


CH 03: VECTORS

PRACTICE: Using the vectors given in the figure, (a) find �⃗⃗� ● �⃗⃗� . (b) Find �⃗⃗� ● �⃗⃗� .

𝒚

𝒙

�⃗⃗� =15 �⃗⃗� =10

�⃗⃗� =20

30°

53°

30°


CH 03: VECTORS

CONCEPT: CALCULATING DOT PRODUCT USING VECTOR COMPONENTS

● You’ll need to calculate the dot product �⃗⃗� ● �⃗⃗� of vectors using unit vector components instead of magnitudes & angles.

- Remember: Dot Product is the multiplication of parallel components, and always results in a number!

EXAMPLE: Calculate 𝐴 ● �⃗� for each of the following:

DOT PRODUCT USING MAG. + ANGLES DOT PRODUCT USING COMPONENTS

�⃗⃗� = 𝐴𝑥 �̂� + 𝐴𝑦𝒋̂ + 𝐴𝑧�̂�

�⃗⃗� = 𝐵𝑥 �̂� + 𝐵𝑦𝒋̂ + 𝐵𝑧�̂�

�⃗⃗� ● �⃗⃗� = _______ + _______ + _______

a) �⃗⃗� = 2𝑖̂ + 3𝑗̂

�⃗⃗� = 𝑖̂ + 2𝑗̂

b) �⃗⃗� = −3𝑖̂ + 𝑗̂ + 4�̂�

�⃗⃗� = 𝑖̂ − 2𝑗̂

�⃗⃗� = 4

�⃗⃗� = 3 60°

�⃗⃗� ● �⃗⃗� = |A| |B| cos𝜃


CH 03: VECTORS

PRACTICE: Calculate the dot product between �⃗⃗� = (𝟔. 𝟔 �̂� − 𝟑. 𝟒 𝒋̂ − 𝟔. 𝟒 �̂�) and �⃗⃗� = (𝟖. 𝟔 �̂� + 𝟐. 𝟔 𝒋̂ − 𝟓. 𝟖 �̂�).

�⃗⃗� ● �⃗⃗� = |A| |B| cos𝜃 �⃗⃗� ● �⃗⃗� = AxBx + AyBy + AzBz

Magnitude & Direction Unit Vector Components


CH 03: VECTORS

EXAMPLE: Vector �⃗⃗� = 7.2�̂� − 3.9𝒋 ̂and �⃗⃗� = 2.1�̂� + 4.8𝒋̂. (a) Calculate �⃗⃗� ● �⃗⃗� . (b) What is the angle between �⃗⃗� & �⃗⃗� ?

�⃗⃗� ● �⃗⃗� = |A| |B| cos𝜃 �⃗⃗� ● �⃗⃗� = AxBx + AyBy + AzBz

Magnitude & Direction Unit Vector Components


CH 03: VECTORS

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