1.25 & Orthogonal vectors: Insights via drawing. (Section 2.10) Consider three unit vectors â, b, and ĉ. Vector â is perpendicular to vector b. Vector b is perpendicular to vector ĉ. Vector a is not parallel to vector ĉ. In all cases, â is perpendicular to ĉ. True/False. Explain your answer by drawing â, b, ĉ and relevant angles.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![(Section 2.9 and Hw 1.18)
be used to show that the magnitude of the vector ca, (e is a
number and a, is a unit vector) can be written solely in terms of e (without a.).
Result:
|câs =
1.20 † Magnitude of the vector v. Show work. (Section 2.9)
- abs(c)
Knowing the angle between a unit vector i and unit vector j is 110°,
calculate a numerical value for the magnitude of ý = 3i + 4J
Result:
v| = 4.098
1.21 Angle between vectors.
Note: The answer is not 5.
(Section 2.9)
Referring to the figure to the right, find the numerical value
for the angle between vector a and vector b.
2(ã, b) =
1.22 Visual estimation of vector dot/cross-products. Show work. (Sections 2.9 and 2.10)
Estimate (e.g., using your pinky) the magnitude of the vector p shown below.
Estimate the angle between p and ğ, p · ġ, and the magnitude of p x q.
Result: (Provide numerical results with 1 or more significant digits).
Note: 1 inch A 2.54 cm.
Show work.
2(p, á) =
cm
cm2
cm?
1.23 4 Form the unit vector û having the same direction as câx. (Section 2.4)
Result:
u =
ax
Note: âx is a unit vector and e is a on-zero real number, e.g., 3 or -3
1.24 4 Coefficient of û in cross products - definitions and trig functions. (Section 2.10)
The cross product of vectors a and b can be written in terms of a real scalar s as axb = sû
where û is a unit vector perpendicular to both a and b in a direction defined by the right-hand
rule. The coefficient s of the unit vector û is inherently non-negative. True/False.
1.25 & Orthogonal vectors: Insights via drawing. (Section 2.10)
Consider three unit vectors â, b, and ĉ.
Vector â is perpendicular to vector b.
Vector b is perpendicular to vector ĉ.
Vector a is not parallel to vector ĉ.
In all cases, â is perpendicular to ĉ. True/False.
Explain your answer by drawing â, b, ĉ and relevant angles.
1.26 Calculating distance between a point and a line via cross-products. (Section 2.10.2)
Draw a horizontally-right unit vector ây and vertically-upward unit vector ấy.
Draw a point Q whose position vector from a point P is ř/P = 5ây.
Draw a line L that passes through point P and is parallel to û =ây + ây.
Calculate the distance d between Q and L using both formulas in equation (2.9)
Result:
(2.9
(2.9)
Homework 1: Vectors - basis independent
60
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