VECTORS

Pythagorean Theorem• Recall that a right triangle has a 90° angle as one of its

angles.• The side that is opposite the 90° angle is called the

hypotenuse.• The theorem due to Pythagoras says that the square of

the hypotenuse is equal to the sum of the squares of the legs. c2 = a2 + b2

a c

b

Trigonometry

Trigonometry is concerned with the connection between the sides and angles in any right angled triangle.

Angle

Introduction to Trigonometry• In this section we define the three basic trigonometric ratios, sine, cosine and tangent.

• opp is the side opposite angle A• adj is the side adjacent to angle A• hyp is the hypotenuse of the right triangle

hyp opp

adj A

There are three formulae involved in trigonometry:

sin A=

cos A=

tan A =

S O H C A H T O A

Using ratios to find angles

We have just found that a scientific calculator holds the ratio information for sine (sin), cosine (cos) and tangent (tan) for all angles.

It can also be used in reverse, finding an angle from a ratio.

To do this we use the sin-1, cos-1 and tan-1 function keys.

Finding an angle from a triangleTo find a missing angle from a right-angled triangle we need to know two of the sides of the triangle.

We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio.

Find angle θ

a) Identify/label the names of the sides.

b) Choose the ratio that contains BOTH of the letters.

14 cm

6 cmθ

1.

Finding a side from a triangle

To find a missing side from a right-angled triangle we need to know one angle and one other side.

Cos45o = 13x

To leave x on its own we need to move the ÷ 13.

It becomes a “times” when it moves.

Note: If

Cos45o x 13 = x

Finding a side from a triangleThere are occasions when the unknown letter is on the bottom of the fraction after substituting.

Cos45o = u13

Move the u term to the other side.

It becomes a “times” when it moves.

Cos45o x u = 13

To leave u on its own, move the cos 45 to other side, it becomes a divide.

u = 45 Cos13

Values of Trigonometric Functions for Common Angles

θ 0o 30o 37o 45o 53o 60o 90o

sinθ 0 0.500 0.600 0.707 0.800 0.866 1.00

cosθ 1.00 0.866 0.800 0.707 0.600 0.500 0

tanθ 0 0.577 0.750 1.00 1.33 1.17

Vector vs. Scalar Review• 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)

Vector Notation• 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

A

Properties of Vectors• 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

More Properties of Vectors• 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•

; 0 A B A A

R A B

Adding Vectors• When adding vectors, their directions must be taken into

account• Units must be the same • Geometric Methods

• Use scale drawings• Algebraic Methods

• 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

A

A

Graphically Adding Vectors, cont.

• 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

A

R

Graphically Adding Vectors, cont.

• 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

Notes about Vector Addition

• Vectors obey the Commutative Law of Addition• The order in which the

vectors are added doesn’t affect the result

• A B B A

Vector Subtraction

• Special case of vector addition• Add the negative of the

subtracted vector• • Continue with standard vector addition procedure

A B A B

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

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

Components of a Vector, cont.• 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,

cosA A x

sinyA A

x y A A A

More About Components of a Vector• 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

More About Components, cont.

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

• May still have to find θ with respect to the positive x-axis• The value will be correct only if the angle lies in the first

or fourth quadrant• In the second or third quadrant, add 180°

2 2 1tan yx y

x

AA A A and

A

A

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

• This gives Rx:

xx vR

Adding Vectors Algebraically, cont.• Add all the y-components

• 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:

yy vR

2y

2x RRR

x

y1

R

Rtan