Scalars and vectors
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Scalars and Vectors. A scalar quantity is one that can be described by a single number: temperature, speed, mass. A vector quantity deals inherently with both magnitude and direction : velocity, force, displacement, acceleration. Scalars and Vectors.

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Scalars and Vectors

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Scalars and vectors

Scalars and Vectors

A scalar quantity is one that can be described

by a single number:

temperature, speed, mass

A vector quantity deals inherently with both

magnitude and direction:

velocity, force, displacement, acceleration

Scalars and vectors1

Scalars and Vectors

Arrows are used to represent vectors. The

direction of the arrow gives the direction of

the vector.

By convention, the length of a vector

arrow is proportional to the magnitude

of the vector.

8 lb

4 lb

Scalars and vectors


The plane travels with a velocity relative to the ground which is vector sum of the plane’s velocity (relative to the air) plus the wind velocity

The resultant is a combination of both motions

Scalars and vectors

Vector Procedures

These are the rules you must play by to add vectors and find the resultant:

Draw each vector to scale and label

Vectors should be drawn in order as they occurred and labeled (I.e.v1, v2…..)

Draw each successive vector starting from the tip of the preceding vector. (This is called the “tip to tail” method

The vector sum is found by drawing the resultant vector arrow from the tail of the first vector to the tip of the last vector

For two-vector operations in two dimensions the resultant forms the diagonal of a parallelogram with the component vectors forming the sides.


Vector v1 is 1 unit east or +1 unit




Vector v2 is 3 units west or - 3 units

Vector (v1 + v2) is 2 units west or - 2 units

R = A1 + A2

v1 + v2 = R



R = Runits @ 0 ( measured from East counterclockwise)

Scalars and vectors

Use a ruler and vector rules to find the resultant of the following vector additions. Record your answer using vector convention.

R = F1+ F2








R = P1+ P2


R = P2+ P1


Vector addition and subtraction

Vector Addition and Subtraction

When a vector is multiplied

by -1, the magnitude of the

vector remains the same, but

the direction of the vector is


Vector addition and subtraction1

Vector Addition and Subtraction

Scalars and vectors

Finding the components of a vector

To find the magnitude of a vector quantity in a particular direction you can use the parallelogram rule

Draw the vector and treat it as the resultant

Form a parallelogram around the resultant in the two directions of interest. In physics problems these are usually perpendicular directions to simplify the problem

Draw the component vectors along the sides of the parallelogram


Pythagorean Theorem:

a2 = ah2 + av2







Pythagorean Theorem:

E2 = Eparalel2 + Eperpendicular2


Scalars and vectors

Using Trig to solve 2D-Vector Problems

Right-Angled Trig Relationships

Trig functions for right-angled triangles are just geometric ratios. You probably memorized these ratios in your math classes using So/h..Ca/h..To/a. Here they are once more!

sin = o / h

cos = a / h



tan = sin / cos = o / a


Follow these rules!

Sketch vectors on a coordinate system. (Two perpendicular axes)

Find (resolve) the components of all vectors in the two directions of interest using the parallelogram rule. Make sure that you use the normal conventions of + and – arithmetic quantities.

Sum the components in the two directions of interest.

Find the magnitude of R using Pythagorean theorem

Use the tangent rule to determine the direction R as measured from the East by convention or a specified direction

Scalars and vectors

Practice, Practice…..Practice

Use trig relationships to determine i) the other component and ii) the resultant R

i) tan = vv / vh



vv = vhtan

vv = (35 m/s) tan 600


vh= 35 m/s

vv = 60.6 m/s

ii) R =  (vv2 + vh2)

R =  ((60.6 m/s)2 + (35 m/s)2)

R = 70 m/s

 = 1800 - 600 = 1200

R = 70 m/s @ 1200

Scalars and vectors

Determine the resultant R of the following vectors by using the horizontal and vertical components of A and B

R = 100 Newtons

A cos 600

A sin 600

A = 80 Newtons


-80 N cos 600

+80 N sin 600

B= 60 Newtons

- 40 N

+ 69.3 N



B cos 300

B sin 300





60 N sin 300

+60 N cos 300



R =  (RH2 + Rv2)

+ 52 N

+ 30 N

R =  ((12 N)2 + (99.3 N)2)

+ 12 N

+ 99.3 N

R = 100 N

 = tan-1 (Rv / RH)

 = tan-1 (99.3 N / 12 N)

R = 100 N @ 830

 = 830

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