3.1 Vectors and Their Properties

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Notes: Vectors

3.1 Vectors and Their Properties

· All physical quantities encountered in this text will be either a scalar or a vector

· A vector quantity has both magnitude (size) and direction

· A scalar is completely specified by only a magnitude (size)

· When handwritten, use an arrow:

· When printed, will be in bold print with an arrow:

· When dealing with just the magnitude of a vector in print, an italic letter will be used: A

Equality of Two Vectors

· Two vectors are equal if they have the same magnitude and the same direction

Movement of vectors in a diagram

· Any vector can be moved parallel to itself without being affected

Negative Vectors

· Two vectors are negative if they have the same magnitude but are 180° apart (opposite directions)

Resultant Vector

· The resultant vector is the sum of a given set of vectors

· When adding vectors, their directions must be taken into account

o Units must be the same

· Geometric Methods

o Use scale drawings

· Algebraic Methods

o More convenient

Adding Vectors Geometrically (Triangle or Polygon Method)

· Choose a scale

· Draw the first vector with the appropriate length and in the direction specified, with respect to a coordinate system

· Draw the next vector with the appropriate length and in the direction specified, with respect to a coordinate system whose origin is the end of vector and parallel to the coordinate system used for

· Continue drawing the vectors “tip-to-tail”

· The resultant is drawn from the origin of to the end of the last vector

· Measure the length of and its angle

· Use the scale factor to convert length to actual magnitude

· When you have many vectors, just keep repeating the process until all are included

· The resultant is still drawn from the origin of the first vector to the end of the last vector

· Vectors obey the Commutative Law of Addition

· The order in which the vectors are added doesn’t affect the result

Vector Subtraction

· Special case of vector addition

· Add the negative of the subtracted vector

· Continue with standard vector addition procedure

Multiplying or Dividing a Vector by a Scalar

· The result of the multiplication or division is a vector

· The magnitude of the vector is multiplied or divided by the scalar

· If the scalar is positive, the direction of the result is the same as of the original vector

· If the scalar is negative, the direction of the result is opposite that of the original vector

Quick Quiz 3.1: The magnitudes of two vectors and are 12 units and 8 units, respectively. What are the largest and smallest possible values for the magnitude of the resultant vector ?

a. 14.4 and 4

b. 12 and 8

c. 20 and 4

d. None of these

Quick Quiz 3.2: If vector is added to vector , the resultant vector has magnitude A + B when and are

a. perpendicular to each other

b. oriented in the same direction

c. oriented in opposite directions

d. none of these answers.

Example 3.1: A car travels 20.0 km due north and then 35.0 km in a direction 60° west of north, as in the diagram. Using a graph, find the magnitude and direction of a single vector that gives the net effect of the car’s trip. This vector is called the car’s resultant displacement.

3.2 Components of a Vector

· A component is a part

· It is useful to use rectangular components

· These are the projections of the vector along the x- and y-axes

· The x-component of a vector is the projection along the x-axis

· The y-component of a vector is the projection along the y-axis

· Then,

· The previous equations are valid only if θ is measured with respect to the x-axis

· The components can be positive or negative and will have the same units as the original vector

· The components are the legs of the right triangle whose hypotenuse is

o May still have to find θ with respect to the positive x-axis

o The value will be correct only if the angle lies in the first or fourth quadrant

o In the second or third quadrant, add 180°

Quick Quiz 3.3: The diagram shows two vectors lying in the xy-plane. Determine the signs of the x- and y-components of , and

Vector / x-component / y-component

Example 3.2: Find the horizontal and vertical components of the 1.00 x 102 m displacement of a superhero who flies from the top of a tall building along the path shown in the diagram. Suppose in stead the superhero leaps in the other direction along a displacement vector, to the top of a flagpole where the displacement components are given by Bx = -25.0 m and By = 10.0 m. Find the magnitude and direction of the displacement vector.

Adding Vectors Algebraically

· Choose a coordinate system and sketch the vectors

· Find the x- and y-components of all the vectors

· Add all the x-components

o This gives Rx:

· Add all the y-components

o This gives Ry:

· Use the Pythagorean Theorem to find the magnitude of the resultant:

· Use the inverse tangent function to find the direction of R:

Interactive Example 3.3: A hiker begins a trip by first walking 25.0 km southeast from her base camp. On the second day she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower.

a. Determine the components of the hiker’s displacements in the first and second days

b. Determine the components of the hiker’s total displacement for the trip.

c. Find the magnitude and direction of the displacement from the base camp.

3.3 Displacement, Velocity and Acceleration in Two Dimensions

Displacement

· Using + or – signs is not always sufficient to fully describe motion in more than one dimension

· Vectors can be used to more fully describe motion

· Still interested in displacement, velocity, and acceleration

· The position of an object is described by its position vector,

· The displacement of the object is defined as the change in its position

Velocity

· The average velocity is the ratio of the displacement to the time interval for the displacement

o The instantaneous velocity is the limit of the average velocity as Δt approaches zero

o The direction of the instantaneous velocity is along a line that is tangent to the path of the particle and in the direction of motion

Acceleration

· The average acceleration is defined as the rate at which the velocity changes

· The instantaneous acceleration is the limit of the average acceleration as Δt approaches zero

Unit Summary (SI)

· Displacement m

· Average velocity and instantaneous velocity m/s

· Average acceleration and instantaneous acceleration m/s2

Quick Quiz 3.4: Which of the following objects can’t be accelerating? (a) an object moving with a constant speed; (b) an object moving with a constant velocity; (c) an object moving along a curve.

Quick Quiz 3.5: Consider the following controls in an automobile: gas pedal, brake, steering wheel. The controls in this list that cause an acceleration of the car are (a) All three (b) the gas pedal and the brake (c) only the brake or (d) only the gas pedal