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!

<< 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

	23 Pesticides/Small animal poison Test By Brooke Delaney Start Exam
	11 Biology 11 Meiosis Sexual Reproduction MCQ By OpenStax Start Quiz
	6 Neuroanatomy 06 Head Somatic Visceral Sensory By Stephen Voron Start Quiz
	43 Biology 43 Animal Reproduction Development MCQ By OpenStax Start Quiz
	2 Neuroscience Exam 2004 3 By David Corey Start Exam
	2 Microeconomics 02 Choice in a World of Scarcity By OpenStax Start Flashcards
	Business Law Ethics By Kevin Moquin Start Quiz
	Fluid Mechanics MCQ By Stephanie Redfern Start Quiz
	4 Practice Neuroanatomy MCQ Exam By David Corey Start Exam
	4 AP 04 Tissue Level of Organization Essay By OpenStax Start Flashcards