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. $\vec{C} = \vec{A} / B$

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

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

They are orthogonal.

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

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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.

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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

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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 $\vec{B} = (−2.0 \hat{i} + 3.0 \hat{j} + 2.0 \hat{k}) m$ .

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

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