2.4 Products of vectors (Page 8/16)

University physics volume 1 Page 8 / 16

Key equations

Multiplication by a scalar (vector equation)	$\vec{B} = α \vec{A}$
Multiplication by a scalar (scalar equation for magnitudes)	$B = \| α \| A$
Resultant of two vectors	${\vec{D}}_{A D} = {\vec{D}}_{A C} + {\vec{D}}_{C D}$
Commutative law	$\vec{A} + \vec{B} = \vec{B} + \vec{A}$
Associative law	$(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$
Distributive law	$α_{1} \vec{A} + α_{2} \vec{A} = (α_{1} + α_{2}) \vec{A}$
The component form of a vector in two dimensions	$\vec{A} = A_{x} \hat{i} + A_{y} \hat{j}$
Scalar components of a vector in two dimensions	${\begin{matrix} A_{x} = x_{e} - x_{b} \\ A_{y} = y_{e} - y_{b} \end{matrix}$
Magnitude of a vector in a plane	$A = \sqrt{A_{x}^{2} + A_{y}^{2}}$
The direction angle of a vector in a plane	$θ_{A} = {tan}^{−1} (\frac{A_{y}}{A_{x}})$
Scalar components of a vector in a plane	${\begin{matrix} A_{x} = A cos θ_{A} \\ A_{y} = A sin θ_{A} \end{matrix}$
Polar coordinates in a plane	${\begin{matrix} x = r cos φ \\ y = r sin φ \end{matrix}$
The component form of a vector in three dimensions	$\vec{A} = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}$
The scalar z -component of a vector in three dimensions	$A_{z} = z_{e} - z_{b}$
Magnitude of a vector in three dimensions	$A = \sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}}$
Distributive property	$α (\vec{A} + \vec{B}) = α \vec{A} + α \vec{B}$
Antiparallel vector to $\vec{A}$	$− \vec{A} = − A_{x} \hat{i} - A_{y} \hat{j} - A_{z} \hat{k}$
Equal vectors	$\vec{A} = \vec{B} \Leftrightarrow {\begin{matrix} A_{x} = B_{x} \\ A_{y} = B_{y} \\ A_{z} = B_{z} \end{matrix}$
Components of the resultant of N vectors	${\begin{matrix} F_{R x} = \sum_{k = 1}^{N} F_{k x} = F_{1 x} + F_{2 x} + … + F_{N x} \\ F_{R y} = \sum_{k = 1}^{N} F_{k y} = F_{1 y} + F_{2 y} + … + F_{N y} \\ F_{R z} = \sum_{k = 1}^{N} F_{k z} = F_{1 z} + F_{2 z} + … + F_{N z} \end{matrix}$
General unit vector	$\hat{V} = \frac{\vec{V}}{V}$
Definition of the scalar product	$\vec{A} \cdot \vec{B} = A B cos φ$
Commutative property of the scalar product	$\vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A}$
Distributive property of the scalar product	$\vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C}$
Scalar product in terms of scalar components of vectors	$\vec{A} \cdot \vec{B} = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}$
Cosine of the angle between two vectors	$cos φ = \frac{\vec{A} \cdot \vec{B}}{A B}$
Dot products of unit vectors	$\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$
Magnitude of the vector product (definition)	$\| \vec{A} \times \vec{B} \| = A B sin φ$
Anticommutative property of the vector product	$\vec{A} \times \vec{B} = − \vec{B} \times \vec{A}$
Distributive property of the vector product	$\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$
Cross products of unit vectors	${\begin{cases} \hat{i} \times \hat{j} = + \hat{k}, \\ \hat{j} \times \hat{k} = + \hat{i}, \\ \hat{k} \times \hat{i} = + \hat{j} . \end{cases}$
The cross product in terms of scalar components of vectors	$\vec{A} \times \vec{B} = (A_{y} B_{z} - A_{z} B_{y}) \hat{i} + (A_{z} B_{x} - A_{x} B_{z}) \hat{j} + (A_{x} B_{y} - A_{y} B_{x}) \hat{k}$

Conceptual questions

What is wrong with the following expressions? How can you correct them? (a) $C = \vec{A} \vec{B}$ , (b) $\vec{C} = \vec{A} \vec{B}$ , (c) $C = \vec{A} \times \vec{B}$ , (d) $C = A \vec{B}$ , (e) $C + 2 \vec{A} = B$ , (f) $\vec{C} = A \times \vec{B}$ , (g) $\vec{A} \cdot \vec{B} = \vec{A} \times \vec{B}$ , (h) $\vec{C} = 2 \vec{A} \cdot \vec{B}$ , (i) $C = \vec{A} / \vec{B}$ , and (j) $C = \vec{A} / B$ .

a. $C = \vec{A} \cdot \vec{B}$ , b. $\vec{C} = \vec{A} \times \vec{B}$ or $\vec{C} = \vec{A} - \vec{B}$ , c. $\vec{C} = \vec{A} \times \vec{B}$ , d. $\vec{C} = A \vec{B}$ , e. $\vec{C} + 2 \vec{A} = \vec{B}$ , f. $\vec{C} = \vec{A} \times \vec{B}$ , g. left side is a scalar and right side is a vector, h. $\vec{C} = 2 \vec{A} \times \vec{B}$ , i. $\vec{C} = \vec{A} / B$ , j. $\vec{C} = \vec{A} / B$

Got questions? Get instant answers now!

If the cross product of two vectors vanishes, what can you say about their directions?

Got questions? Get instant answers now!

If the dot product of two vectors vanishes, what can you say about their directions?

They are orthogonal.

Got questions? Get instant answers now!

What is the dot product of a vector with the cross product that this vector has with another vector?

Got questions? Get instant answers now!

Problems

Assuming the + x -axis is horizontal to the right for the vectors in the following figure, find the following scalar products: (a) $\vec{A} \cdot \vec{C}$ , (b) $\vec{A} \cdot \vec{F}$ , (c) $\vec{D} \cdot \vec{C}$ , (d) $\vec{A} \cdot (\vec{F} + 2 \vec{C})$ , (e) $\hat{i} \cdot \vec{B}$ , (f) $\hat{j} \cdot \vec{B}$ , (g) $(3 \hat{i} - \hat{j}) \cdot \vec{B}$ , and (h) $\hat{B} \cdot \vec{B}$ .

The x y coordinate system has positive x to the right and positive y up. Vector A has magnitude 10.0 and points 30 degrees counterclockwise from the positive x direction. Vector B has magnitude 5.0 and points 53 degrees counterclockwise from the positive x direction. Vector C has magnitude 12.0 and points 60 degrees clockwise from the positive x direction. Vector D has magnitude 20.0 and points 37 degrees clockwise from the negative x direction. Vector F has magnitude 20.0 and points 30 degrees counterclockwise from the negative x direction.

Got questions? Get instant answers now!

Assuming the + x -axis is horizontal to the right for the vectors in the preceding figure, find (a) the component of vector $\vec{A}$ along vector $\vec{C}$ , (b) the component of vector $\vec{C}$ along vector $\vec{A}$ , (c) the component of vector $\hat{i}$ along vector $\vec{F}$ , and (d) the component of vector $\vec{F}$ along vector $\hat{i}$ .

a. 8.66, b. 10.39, c. 0.866, d. 17.32

Got questions? Get instant answers now!

Find the angle between vectors for (a) $\vec{D} = (−3.0 \hat{i} - 4.0 \hat{j}) m$ and $\vec{A} = (−3.0 \hat{i} + 4.0 \hat{j}) m$ and (b) $\vec{D} = (2.0 \hat{i} - 4.0 \hat{j} + \hat{k}) m$ and $\vec{B} = (−2.0 \hat{i} + 3.0 \hat{j} + 2.0 \hat{k}) m$ .

Got questions? Get instant answers now!

Questions & Answers

how do you get the 2/50

Abba Reply

number of sport play by 50 student construct discrete data

Aminu Reply

width of the frangebany leaves on how to write a introduction

Theresa Reply

Solve the mean of variance

Veronica Reply

Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ... Step 2: Find each score's deviation from the mean. ... Step 3: Square each deviation from the mean. ... Step 4: Find the sum of squares. ... Step 5: Divide the sum of squares by n – 1 or N.

kenneth

what is error

Yakuba Reply

Is mistake done to something

Vutshila

anas

What is the life teble

anas

Jibrin

statistics is the analyzing of data

Tajudeen Reply

what is statics?

Zelalem Reply

how do you calculate mean

Gloria Reply

diveving the sum if all values

Shaynaynay

let A1,A2 and A3 events be independent,show that (A1)^c, (A2)^c and (A3)^c are independent?

Fisaye Reply

what is statistics

Akhisani Reply

data collected all over the world

Shaynaynay

construct a less than and more than table

Imad Reply

The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.

Aschalew Reply

Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400 a. what is the probability of getting more than 12,000 hits? b. what is the probability of getting fewer than 9,000 hits?

Akshay Reply

Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400. a. What is the probability of getting more than 12,000 hits

Akshay

Bright

Sorry i want to learn more about this question

Bright

Someone help

Bright

a= 0.20233 b=0.3384

Sufiyan

Shaynaynay

How do I interpret level of significance?

Mohd Reply

It depends on your business problem or in Machine Learning you could use ROC- AUC cruve to decide the threshold value

Shivam

how skewness and kurtosis are used in statistics

Owen Reply

yes what is it

Taneeya

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 6

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?

Ask

	1 Physiotherapy Flashcards Set 1 By Rhodes Start Flashcards
	Art Reviewer MCQ By Edgar Delgado Start Quiz
	Anthropology Marriage Family Household By Richley Crapo Start Assignment
	Spanish (Places) By Inderjeet Brar Start Flashcards
	Cultural Anthropology Definition By Richley Crapo Start Assignment
©flickr:	Core Java Unit Test-1(CAJ003) By Abishek Devaraj Start Quiz
	4 Practice Neuroanatomy MCQ Exam By David Corey Start Exam
	9 Arts Society: Theater 9 Exam By Jonathan Long Start Exam
	3 Sociology 03 Culture MCQ By OpenStax Start Quiz
	MAPEH TEST By Edgar Delgado Start Test