Chapter 3 Vectors and Two-Dimensional Motion

Vectors and Scalars A scalar is a quantity that is completely specified by a positive or negative number with an appropriate unit and has no direction. A scalar is a quantity that is completely specified by a positive or negative number with an appropriate unit and has no direction. A vector is a physical quantity that must be described by a magnitude (number) and appropriate units plus a direction. A vector is a physical quantity that must be described by a magnitude (number) and appropriate units plus a direction.

Scalar Quantity Scalar Quantity a quantity that has magnitude but not direction a quantity that has magnitude but not direction Vector Quantity Vector Quantity a quantity that has both magnitude and direction a quantity that has both magnitude and direction

Magnitude – the numerical value of a scalar or vector. Magnitude – the numerical value of a scalar or vector. For example, a velocity vector might be 30 m/s to the west. The magnitude of this vector is 30 m/s. A force vector might be 100 pounds upward. The magnitude of this vector is 100 pounds.

Scalar Quantities length length e.g. 93,000,000 miles e.g. 93,000,000 miles mass mass e.g. 180 kg e.g. 180 kg speed speed e.g. 186,000 miles/second e.g. 186,000 miles/second

More examples More examples Temperature (20 o C) Temperature (20 o C) Volume (45 cm 3) Volume (45 cm 3) Time intervals (24 h) Time intervals (24 h) Rules of ordinary arithmetic are used to manipulate scalar quantities Rules of ordinary arithmetic are used to manipulate scalar quantities

Vector Quantities force force e.g. 20 Newtons Eastward e.g. 20 Newtons Eastward velocity velocity e.g. 20 meters/second North e.g. 20 meters/second North acceleration acceleration e.g. 9.8 m/s 2 downward e.g. 9.8 m/s 2 downward

Vectors are used to denote quantities that have magnitude and direction are used to denote quantities that have magnitude and direction can be added and subtracted can be added and subtracted can be multiplied or divided by a number can be multiplied or divided by a number can be manipulated graphically (i.e., by drawing them out) or algebraically (by considering components) can be manipulated graphically (i.e., by drawing them out) or algebraically (by considering components)

Vector an arrow drawn to scale is used to represent a vector quantity an arrow drawn to scale is used to represent a vector quantity vector notation vector notation F

Vector Notation When handwritten, use an arrow: When handwritten, use an arrow: When printed, will be in bold print: A When printed, will be in bold print: A

Vector Example A particle travels from A to B along the path shown by the dotted red line A particle travels from A to B along the path shown by the dotted red line This is the distance traveled and is a scalar This is the distance traveled and is a scalar The displacement is the solid line from A to B The displacement is the solid line from A to B The displacement is independent of the path taken between the two points The displacement is independent of the path taken between the two points Displacement is a vector Displacement is a vector Notice the arrow indicating direction Notice the arrow indicating direction

Properties of Vectors Equality of Two Vectors Equality of Two Vectors Two vectors are equal if they have the same magnitude and the same direction Two vectors are equal if they have the same magnitude and the same direction Movement of vectors in a diagram Movement of vectors in a diagram Any vector can be moved parallel to itself without being affected Any vector can be moved parallel to itself without being affected

Equivalent Vectors Vectors are said to be equivalent if both their magnitudes and direction match

Adding Vectors When adding vectors, their directions must be taken into account When adding vectors, their directions must be taken into account Units must be the same Units must be the same Graphical Methods Graphical Methods Use scale drawings Use scale drawings Accuracy difficult to control Accuracy difficult to control Algebraic Methods Algebraic Methods Accuracy well defined Accuracy well defined

More Properties of Vectors Resultant Vector Resultant Vector The resultant vector is the sum of a given set of vectors The resultant vector is the sum of a given set of vectors

Graphical Methods of Vector Addition Graphical Methods of Vector Addition tip-to-tail method tip-to-tail method parallelogram method parallelogram method

The Tip-to-Tail Method

To add vector B to vector A: Draw vector A. Draw vector A. Draw vector B with its tail starting from the tip of A. Draw vector B with its tail starting from the tip of A. The sum vector A+B is the vector drawn from the tail of vector A to the tip of vector B. The sum vector A+B is the vector drawn from the tail of vector A to the tip of vector B.

A B C D E F (A = 4 cm; B = 2 cm; C = 3 cm; D = 2 cm; E = 3 cm; F = 2 cm)

A + B = ? B A A + B

A D A + D = ? A + D

C D C + D = ? C + D

B E B + E = ? B + E

A F A + F = ? A + F

Adding Vectors Graphically, cont. When you have many vectors, just keep repeating the process until all are included 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 The resultant is still drawn from the origin of the first vector to the end of the last vector

A B D A + B + D = ? A + B + D

The order in which the vectors are added does not matter.

A B D D + B + A = ? D + B + A A B D A + B + D

A B C D F A + B + D + C + F = ? A + B + D + C + F

Why we need to use vectors? Riverboat: V VCVC V=Velocity of boat in calm water. V C = Velocity of Current V R = Resulting Velocity VRVR V

Vector vs. Scalar Review A vector quantity has both magnitude (size) and direction A vector quantity has both magnitude (size) and direction A scalar is completely specified by only a magnitude (size) A scalar is completely specified by only a magnitude (size)

Alternative Graphical Method When you have only two vectors, you may use the Parallelogram Method When you have only two vectors, you may use the Parallelogram Method All vectors, including the resultant, are drawn from a common origin All vectors, including the resultant, are drawn from a common origin The remaining sides of the parallelogram are sketched to determine the diagonal, R The remaining sides of the parallelogram are sketched to determine the diagonal, R A + B = ? B A A + B

Multiplying or Dividing a Vector by a Scalar The result of the multiplication or division is a vector The result of the multiplication or division is a vector The magnitude of the vector is multiplied or divided by the scalar 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 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 If the scalar is negative, the direction of the result is opposite that of the original vector

Find the vector “B – A”

Two unknown vectors A and B are added. The magnitude of the sum vector “A + B” (i.e., the quantity |A + B|) 1.is at least as great as |A| (i.e., the magnitude of A). 2.is at most as great as |A| + |B| (i.e., the magnitudes of A and B added). 3.must be equal to |A| + |B|. 4.can be greater than |A| + |B|.