CHAPTER 3

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Vectors

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Vectors and Scalars,Addition of vectorsSubtraction of vectors

x

y

r

Physics deals with many quantities that have both

MagnitudeDirection

VECTORS !!!!!

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A scalar quantity is a quantity that has magnitude only and has no direction in space

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Examples of Scalar Quantities:

Length Area Volume Time Mass

Scalar

A vector quantity is a quantity that has both magnitude and a direction in space

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Examples of Vector Quantities: Displacement Velocity Acceleration Force

Vector

A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector.

Two vectors are equal if they have the same direction and magnitude (length).

Blue and orange vectors have same magnitude but different direction.

Blue and green vectors have same direction but different magnitude.

Blue and purple vectors have same magnitude and direction so they are equal.

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Examples A = 20 m/s at 35° NE B = 120 lb at 60° SE

C = 5.8 mph/s west

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Example

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•The direction of the vector is 55° North of East

•The magnitude of the vector is 2.3.

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Try Again

Direction:

Magnitude:

43° East of South

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Try Again

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It is also possible to describe this vector's direction as 47 South of East.

Why?

Vector Addition vectors may be added graphically or analytically

Triangle (Head-to-Tail) Method1. Draw the first vector with the proper length and orientation.

2. Draw the second vector with the proper length and orientation originating from the head of the first vector.3. The resultant vector is the vector originating at the tail of the first vector and terminating at the head of the second vector.

4. Measure the length and orientation angle of the resultant. 11

Adding vectors in same direction:Example: Travel 8 km East on day 1, 6 km

East on day 2. Displacement = 8 km + 6 km = 14 km East Example: Travel 8 km East on day 1, 6 km

West on day 2. Displacement = 8 km - 6 km = 2 km East

“Resultant” = Displacement

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13

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Adding more than two vectors graphically

Subtraction of Vectors -graphically

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bad

ba c whereas)b(ad

Parallelogram (Tail-to-Tail) Method1. Draw both vectors with proper length and orientation originating from the same point.2. Complete a parallelogram using the two vectors as two of the sides.3. Draw the resultant vector as the diagonal originating from the tails.4. Measure the length and angle of the resultant vector.

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Components of Force:

x

y

Resolving a Vector Into Components

+x

+y

A

Ax

Ayq

The horizontal, or x-component, of A is found by Ax = A cos q.

The vertical, ory-component, of A is found by Ay = A sin qBy the Pythagorean Theorem, Ax

2 + Ay2 = A2

Every vector can be resolved using these formulas, such that A is the magnitude of A, andq is the angle the vector makes with the x-axis.

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Analytical Method of Vector Addition1. Find the x- and y-components of each vector.

Ax = A cos q Ay = A sin q Bx = B cos q By = B sin q Cx = C cos q Cy = C sin q

2. Sum the x-components. This is the x-component of the resultant.

Rx

3. Sum the y-components. This is the y-component of the resultant.

Ry

4. Use the Pythagorean Theorem to find the magnitude of the resultant vector.Rx

2 + Ry2 = R2

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5. Find the reference angle by taking the inverse tangent of the absolute value of the y-component divided by the x-component.

q = Tan-1 Ry/Rx

6. Use the “signs” of Rx and Ry to determine the quadrant. NE

(+,+)NW

(-,+)

SW(-,-)

SE(-,+)

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Step 1 Sketch the given vector with the tail

located at the origin of an x-y coordinate system. (Ex. 25 m at an angle of 36º)

25 m

36º

Step 2 Draw a line segment from the tip of the

vector perpendicular to the x-axis

25 m

Notice, you now have a right triangle with a known hypotenuse and known angle measurements

36º

Step 3 Replace the perpendicular sides of the

right triangle with vectors drawn tip – to - tail

25 m

Step 4 Use sine and cosine functions to find the

horizontal and vertical components of the given vector.

25 m

Rx

Ry36º

Cos(36) = Rx/25

Rx = 25cos(36)

Rx = 20.2 m

Sin(36) = Ry/25

Ry = 25sin(36)

Ry = 14.7 m

Example:

5 N6 N

x y5 cos 30° = +4.33 5 sin 30° = +2.5

6 cos 45 ° = - 4.24

6 sin 45 ° = + 4.24

+ 0.09 + 6.74

R = (0.09)2 + (6.74)2 = 6.74 N

q = arctan 6.74/0.09 = 89.2°

135°45° 30°

R

Rx

Ry

Problem 1

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Problem 2

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Problem 3

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Problem 4

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