REVIEW OF VECTORS AND TRIGONOMETRY F. W. ADAM MECHANICAL ENGINEERING DEPARTMENT KNUST JULY 2013
Dec 31, 2015
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