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13: Vector Functions (Exercises) - Mathematics

13: Vector Functions (Exercises) - Mathematics



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These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.

13.2: Calculus with Vector Functions

13.3: Arc length and Curvature

13.4: Motion Along a Curve


MATLAB Lesson 3 - Vectors

The standard vector operations of adding two vectors and multiplying a vector by a scalar work in MATLAB.

However the straight forward multiplication or division of vectors is not defined.

Addition of two vectors

The vector u has 3 elements 1, 2, 3.

The vector v has 3 elements 10, 11, 12.

The vector w has 3 elements 11, 13, 15.

MATLAB can handle vectors with any number of elements, even hundreds of thousands of elements. However both vectors must have the same number of elements for their sum to be defined.

The vector u has 5 elements 1, 2, 3.

The vector v has 6 elements 10, 11, 12, 13.

. Error using ==> plus
Matrix dimensions must agree.

A vector times a scalar

Multiplying a vector by a scalar produces another vector of the same size in which each element of the original vector has been multiplied by the scalar.

The vector u has 3 elements 1, 2, 3 from before, so the vector w has elements -2, -4, -6


MATHEMATICS T STPM

SOLUTION FOR QUICK CHECK 6.1

SOLUTION FOR QUICK CHECK 6.2

SOLUTION FOR QUICK CHECK 6.3

SOLUTION FOR QUICK CHECK 6.4

SOLUTION FOR QUICK CHECK 6.5

SOLUTION FOR QUICK CHECK 6.6

SOLUTION FOR QUICK CHECK 6.7

SOLUTION FOR EXERCISE 6.10

SOLUTION FOR EXERCISE 6.11


Theory. Magnitude of a vector

The magnitude of the vector AB is denoted as | AB |.

The magnitude of the vector in Cartesian coordinates is the square root of the sum of the squares of it coordinates.

For example, for the vector a = <>x ay az> magnitude of a vector can be found using the following formula:

You can input only integer numbers, decimals or fractions in this online calculator (-2.4, 5/7, . ). More in-depth information read at these rules.


A subgroup is normal if and only if it is the kernel of a homomorphism.

We are going tho show that the kernel of a group homomorphism is a normal subgroup. Next, we prove that every normal subgroup is the kernel of a (. ) Read More »

How to write matrices in Latex ? matrix, pmatrix, bmatrix, vmatrix, Vmatrix

How to write matrices in Latex ? matrix, pmatrix, bmatrix, vmatrix, Vmatrix. Here are few examples to write quickly (. ) Read More »

Sine is an odd function sin(-x)=-sin x

We prove here that the sine function sin (-x) = - sin x is odd using the unit circle. Read More »

Cosine is even function cos(-x)=cos x

We prove here that the cosine function cos(-x)=cos x is even using the unit circle. Read More »

Newton’s interpolation polynomial

In this section, we shall study the polynomial interpolation in the form of Newton. Given a sequence of (n+1) data points and a function f, the (. ) Read More »

Chain rule proof - derivative of a composite function

Derivative of composite function (g ∘ f) (g circle f), chain rule is defined by (g ∘ f)’(x) = g’(f(x)) × f’(x) . Derivative of composite function (u ∘ (. ) Read More »

Latex imaginary part symbol

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Latex real part symbol

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Latex tensor product

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Latex backslash symbol

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Latex convolution symbol

How to write convolution symbol using Latex ? In function analysis, the convolution of f and g f∗g is defined as the integral of the product of (. ) Read More »

Chebyshev polynomials

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Latex jacobian symbol

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Latex gradient symbol

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Derivative of inverse functions

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Derivative of exp x, e^x

Derivative f’ of the function f(x)=exp x is: f’(x) = exp x for any value of x. Read More »

Derivative of sin x

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Derivative of tan x

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13: Vector Functions (Exercises) - Mathematics

SUBJECT: Mathematics
TOPIC: Vector Addition
DESCRIPTION: A set of problems dealing with vector addition.
CONTRIBUTED BY: Carol Hodanbosi
EDITED BY: Jonathan G. Fairman - August 1996

Purpose:

A vector is a quantity that has both magnitude, or size, and direction. Forces can be represented by vectors, since they have both a size and direction of action. Below is an example of a mass supported by two cables. If the mass is not moving, all the forces acting on the mass are considered to be balanced. You will investigate each of the forces acting on the mass and compare their relationships.


Fw represents the weight of the object. It is found by mulitplying its mass by gravity. That is, Fw = m * g, where g equals 9.8 m/s 2 . This force is directed downward.

Exercises:

  1. If the object represented by F has a mass of 4.8 kilograms, find its weight.
    (answer)

Since vectors CB and CA are not acting parallel or perpendicular to the base of the stand, it is helpful to find the components of each of these vectors. Components are vectors that combine vectorally to form the resultant vector, in this case CB or CA . For example, to find the components of CA or CB one first needs to find the angle that vector forms with the horizontal line, angle ACD or angle BCE , see diagram below.

To find components of vector CB , form a right triangle with CB as the hypotenuse. Since CB is a vector, or a ray, one will be selecting a fixed portion of CB . Recall the trigonometric functions of the sine (side opposite/ hypotenuse) and the cosine (side adjacent / hypotenuse). The sine of angle BCE would equal side BE/CB , while the sine of angle ACD = AD/ AC .
Let's assume that the angle ACD and angle BCE are both 35° and that the weight represented by vector CF is 100 newtons. Since the weight is static, and not moving, we can assume all the forces are balanced. The vector represented by CE (to the right) must be balanced by the vector CD (to the left).The downward force of the weight represented by CF must be balanced by the two upward forces DA and EB . Since the two right triangles have two congruent angles (35°) and two congruent sides, ( FD and CE ) the two triangle are congruent (Leg, Acute angle).
Because DA + EB = CF then DA = EB = 50 newtons.

By substitution, Sin angle BCE = BE/CB


Sin 35° = 50/ CB
CB = 50 /Sin 35°
CB = 87.17 newtons

One can also find the measure of CE or CD using the tangent function.

tan BCE = BE/EC
tan 35° = 50 / EC
EC = 50 /tan 35°
EC = 71. 4 newtons

Now use the following diagram to solve the problems below.

A load of 500 kg is suspended at the end of a horizontal boom supported by a cable. The cable makes a 42° angle with the boom and is attached to a wall by a supporting pin. You can assume the boom's mass is negligible.


13: Vector Functions (Exercises) - Mathematics

A 2D vector is a vector of the vector. Like 2D arrays, we can declare and assign values to a 2D vector!
Assuming you are familiar with a normal vector in C++, with the help of an example we demonstrate how a 2D vector differs from a normal vector below:

In a 2D vector, every element is a vector.

Another approach to access the vector elements:

Like Java’s jagged arrays, each element of a 2D vector can contain a different number of values.

Exercise Problem: Define the 2D vector with different sizes of columns.
Examples:

2D vectors are often treated as a matrix with “rows” and “columns” inside it. Under the hood they are actually elements of the 2D vector.
We first declare an integer variable named “row” and then an array named “column” which is going to hold the value of the size of each row.

After that we proceed to initialize the memory of every row by the size of column.

Another Approach
Suppose we want to initialize a 2D vector of “n” rows and “m” columns, with a value 0.

Yet Another Approach:
Suppose we want to create a 2D vector of “n” rows and “m” columns and input values.

We hope you that you leave this article with a better understanding of 2D vectors and are now confident enough to apply them on your own.


Chapter 10 Class 12 Vector Algebra

Learn Chapter 10 Class 12 Vector Algebra free with solutions of all NCERT Questions, Examples as well as Supplementary Questions from NCERT.

Suppose we have to go 10km from Point A to Point B.

This 10km is the distance travelled.

It is only value - 10, nothing else.

This is a scalar quantity.

We go 10 km East from Point A to Point B.

So, we have travelled 10 km East

It has value - 10, and direction - East.

So, this is a vector quantity.

In this chapter, we will talk about vectors. The topics include

  • Basics - Difference between scalars and vectors, with example
  • Graphical Displacement - Where we represent Displacement like 40 km, 30° East of North, in the Graph.
  • We also discuss what is a position vector, and different types of vectors
  • Then we learn what happens when a scalar is multiplied to a vector
  • We learn what are Equal Vectors and Unit Vectors, and do some questions on them
  • Finding Direction Ratios and Direction Cosines
  • Then, adding two vectors
  • Finding vector joining two points
  • Section formula in vectors
  • Checking if 3 points make a right angled triangle
  • Checking if 2 vectors are collinear
  • Checking if 3 points (or vectors) are collinear
  • Then, Scalar Product of Vectors and its properties
  • Finding projection using scalar product
  • Vector product of vectors, and its properties
  • Finding area using vector product (Area of parallelogram = Vector Product)

All the questions are solved, with step by step explanation. Click on an exercise or a topic to start learning


Position Vector

In these lessons, we will learn what a position vector is and how to find a position vector for a vector between two points.

What Is A Position Vector?

A vector that starts from the origin (O) is called a position vector .

In the following diagram, point A has the position vector a and point B has the position vector b.


Example:

Tutorial On Vectors

What are 2-dimensional Vectors, 3-dimensional Vectors, Displacement Vectors and Position Vectors?

How To Find The Position Vector?

Example:
P is the point (3, 4). (overrightarrow = left( <egin<*<20>>< - 6>3end> ight)).
Find the position vector of Q.

Vectors In R 2 And R 3

Position Vector and Magnitude / Length.
How to find a position vector for a vector between two points and also find the length of the vector?

Example:
a) Find the position vector v for a vector that starts at Q(3, 7) and ends at P(-4, 2)
b) Find the length of the vector found in part a)

How To Find The Position Vector Between Two Points?

Example:
Find the position vector between the point A(3, 2) and the point B(-2, 1)

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.


13: Vector Functions (Exercises) - Mathematics

By looking at the graph we can solve the equation sin(x) = x/3. The roots are determined by the places where the two curves cross.

We can graph surfaces using the plot3d command. For example, to graph z = xy, where x and y run from -1 to 1, we do this:

For more information use the ?plot3d command. No semicolon needed for this one.

Note that there are three solutions, two of which are complex. Now let's try something more complicated:

Note: There is a good reason why we wrote "1.5 " instead of "(3/2)" as the coefficient of x in the last equation. If one of the numbers in the equation is in decimal form, then Maple tries to find an approximate solution in decimal form. If none of the numbers are in decimal form, as in the first example, then Maple tries to find an exact solution. This may fail, since there is no algebraic formula for the roots polynomial equations of degree five or more ( Galois ).

These can contain literal as well as numerical coefficients:

In the second example we have to tell Maple that x and y are the variables to be solved for. Otherwise it wouldn't know.

You can solve systems of two equations in two unknowns of the form f(x) = 0, g(x) = 0 by graphing the functions f(x) and g(x) and seeing where the curves cross.

Maple does arithmetic pretty much as you would expect it to:

It has built-in commands which can do a lot of work quickly. For example, to add up the numbers 1, 1/2, 1/3, . 1/10, we do this:

Note that Maple gave us the exact answer as a fraction in lowest terms. For an approximate answer in decimal form, do this:

The only difference was the 1.0 in place of 1 . Note the decimal point. We can also things like factor numbers:

Note two things. Sometimes we need to use the evalf function to convert results from exact to floating point (decimal) form. Sometimes it is convenient to use the quote(") sign: it stands for the result of the preceding computation.

The last computation deserves comment. Suppose we just do the obvious thing (try it!).

Maple does not give us a numerical answer because the integral of this function cannot be expressed in terms of elementary functions. In particular it cannot be integrated by the usual techniques. However, note that we have surrounded our computation with evalf( . ) . This forces Maple to evaluate the integral numerically: evalf stands for evaluate in floating point form

Here are some examples of matrix calculations. Be sure to use with(linalg) before doing them. Also be sure to try these out to see how they work!

Did you notice the difference between the product ab and the product ba ?

For information and examples on a particular Maple " function", use the "?" command. For example,

gives information on the solve command. Often it is helpful to scroll to the end of the help window and look at the examples, bypassing the technical discussion that precedes it. You can also try the command with no keyword:

This gives additonal information on how to use the help system.

Back to Maple
Back to Department of Mathematics, University of Utah
Last modified by jac March 27, 1995
Copyright © 1995 Department of Mathematics, University of Utah


Watch the video: : Vector Functions u0026 Space Curves (August 2022).