Two Unit Vectors Added

Q. î and ĵ are unit vectors along x- and y- axis respectively. What is the magnitude of the vector î + ĵ? expects 1.41

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Q. î and ĵ are unit vectors $A = 1$ , $B = 1$ along x- and y- axis $φ = 90 °$ respectively. What is the magnitude of the vector î + ĵ?

The playground

Outputs

size

R

The solution

Solved with Vectors

Given

$A = 1$ from “unit vectors”
$B = 1$ from “unit vectors”
$φ = 90 °$ from “along x- and y- axis”

Find $R$ .

Working

law of cosines $R^{2} = A^{2} + B^{2} + 2 \cdot A \cdot B \cdot \cos φ$
$R^{2} = 1^{2} + 1^{2} + 2 \cdot 1 \cdot 1 \cdot \cos 90 °$
$R = 1.41421356$
$R \approx 1.41$ to 3 significant figures

Conclusion

The two unit vectors are at right angles, so their sum is the diagonal of a unit square, longer than either but shorter than the two laid end to end.

The Ruby

1require_relative "../physics"23question :two_unit_vectors_added do4  kind :numerical5  describes "î and ĵ are unit vectors along x- and y- axis respectively. %{size}?"6  asks size: "What is the magnitude of the vector î + ĵ"7end

1require_relative "../questions/two_unit_vectors_added"2require_relative "../science/motion_in_a_plane/vectors"34solve :two_unit_vectors_added, using: Vectors do5  deduce_from "unit vectors", A: 1, B: 16  deduce_from "along x- and y- axis", phi: "90 deg"7  take size: :R8  answer find_value(:size, figures: 3)910  conclusion "The two unit vectors are at right angles, so their sum is the diagonal of a unit square, longer " \11             "than either but shorter than the two laid end to end."12end

1require_relative "../solutions/two_unit_vectors_added"23play_with :two_unit_vectors_added do4  title "Two Unit Vectors Added"5end

The laws

quantity	written	units	within	from
volume	$volume$	—	—	Vectors
mass	$mass$	—	—	Vectors
speed	$speed$	—	—	Vectors
density	$density$	—	—	Vectors
number of moles	$n$	—	—	Vectors
angular frequency	${angular}_{frequency}$	—	—	Vectors
work	$work$	—	—	Vectors
current	$current$	—	—	Vectors
temperature	$temperature$	—	—	Vectors
pressure	$pressure$	—	—	Vectors
time	$time$	—	—	Vectors
power	$power$	—	—	Vectors
total path length	$s$	—	—	Vectors
energy	$energy$	—	—	Vectors
gravitational potential	${gravitational}_{potential}$	—	—	Vectors
coefficient of friction	$μ$	—	—	Vectors
charge	$charge$	—	—	Vectors
acceleration	$acceleration$	—	—	Vectors
velocity	$velocity$	—	—	Vectors
displacement	$displacement$	—	—	Vectors
angular velocity	${angular}_{velocity}$	—	—	Vectors
force	$force$	—	—	Vectors
angular momentum	${angular}_{momentum}$	—	—	Vectors
linear momentum	$p_{lin}$	—	—	Vectors
electric field	${electric}_{field}$	—	—	Vectors
average velocity	${average}_{velocity}$	—	—	Vectors
magnetic moment	${magnetic}_{moment}$	—	—	Vectors
relative velocity	$v_{AB}$	—	—	Vectors
impulse	$impulse$	—	—	Vectors
first vector	$A$	—	0 … ∞	Vectors
second vector	$B$	—	0 … ∞	Vectors
angle between	$φ$	—	0 … 3.142	Vectors
resultant	$R$	—	0 … ∞	Vectors
resultant direction	$β$	—	0 … 3.142	Vectors
direction	$α$	—	-3.142 … 3.142	Vectors
x component	$A_{x}$	—	—	Vectors
y component	$A_{y}$	—	—	Vectors

A = \sqrt{{A_{x}}^{2} + {A_{y}}^{2}}

magnitude from components

R^{2} = A^{2} + B^{2} + 2 \cdot A \cdot B \cdot \cos φ

law of cosines

R \cdot \sin β = B \cdot \sin φ

law of sines

R \cdot \cos β = A + B \cdot \cos φ

along the first

A_{x} = A \cdot \cos α

resolved along x

A_{y} = A \cdot \sin α

resolved along y