review of vectors and trigonometry
DESCRIPTION
mechanics of machinesTRANSCRIPT
REVIEW OF VECTORS AND TRIGONOMETRY
F. W. ADAM
MECHANICAL ENGINEERING DEPARTMENT
KNUSTJULY 2013
REVIEW OF TRIGONOMETRY
• You must have mastered right-triangle trigonometry.
y
x
R
q
siny
R
cosx
R
tany
x
R2 = x2 + y2R2 = x2 + y2
cosec θ = 1/sin θ
secan θ = 1/cos θ
cotan θ = 1/tan θ
• 1 radian = 180°/ π = 57.29577 95130 8232. . . • 1 = π /180 radians = 0.01745 32925 radians
The arc s described when the line ON rotates through is i.e. ⇒
RELATIONSHIPS AMONG TRIGONOMETRIC FUNCTIONS
+ =11+ = 1+ co =
ADDITION FORMULAS
SMALL ANGLES
If is small;
sin tan and cos 1
SSINE AND COSINE RULES
SINE RULE
REVIEW OF VECTORS• A vector is a quantity that has both direction and magnitude. NOTATIONVector quantities are printed in boldface type, and scalar quantities appear in lightface italic type. Thus , the vector quantity V has a scalar V. In long hand work vector quantities should always be consistently indicated by a symbol such as V or to distinguish them from scalar quantities.
Addition P+Q=R
Parallelogram addition
Commutative law P+Q=Q+P Associative law P+(Q+R)=(P+R)+Q
Subtraction
P-Q=P+(-Q)
VECTOR DECOMPOSITION
Unit vectors i, j, k
|i|=|j|=|k|=1
k
|𝑽|=𝑉=√𝑉 𝑥2+𝑉 𝑦
2+𝑉 𝑧2
direction cosines l, m, n are the direction cosines of the angles between V and the x, y, z-axes. Thus,
l=/V m=/V n=/V
So that V=V(lik)
EXAMPLE
Determine the rectangular representation of the 200 N force, F,
-10 N
ExampleA=8i-3j-5k and B=4i-6j+5kA.B=(8i-3j-5k).(4i-6j+5k)
=32+18-25= 25
DOT/SCALAR PRODUCTS
= +
𝑷 .𝑸=|𝑃|∨𝑄∨𝑐𝑜𝑠𝜃
Also
CROSS/VECTOR PRODUCTS
i
kj
𝑷×𝑸=|𝑃||𝑄|𝑠𝑖𝑛𝜃𝒏
𝑸×𝑷=−𝑷×𝑸
R
𝑎𝑛𝑑 𝒊× 𝒊= 𝒋 × 𝒋=𝒌×𝒌=𝟎
CROSS/VECTOR PRODUCTS cont’d…
CROSS/VECTOR PRODUCTS cont’d…
Alternatively
THANK YOU