Two Unit Vectors Added: Heading

solved Vectors

Q. î and ĵ are unit vectors $A = 1$ , $B = 1$ along x- and y- axis $φ = 90 °$ respectively. What angle does the vector î + ĵ make with the x-axis, in degrees?

Outputs

direction

β

How it's solved

Solved with Vectors

Given

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

Find $β$ .

Working

law of cosines $R^{2} = A^{2} + B^{2} + 2 \cdot A \cdot B \cdot \cos φ$
$R^{2} = 1^{2} + 1^{2} + 2 \cdot 1 \cdot 1 \cdot \cos 90 °$
$R = 1.41421356$
law of sines $R \cdot \sin β = B \cdot \sin φ$
$1.41421356 \cdot \sin β = 1 \cdot \sin 90 °$
$β = 45.00°$

Conclusion

The two parts are equal, so their sum runs along the diagonal of a square, halfway between the two axes.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :two_unit_vectors_added_heading do4  kind :numerical5  describes "î and ĵ are unit vectors along x- and y- axis respectively. %{direction}, in degrees?"6  asks direction: "What angle does the vector î + ĵ make with the x-axis"7end

solution.rb

1require_relative "../questions/two_unit_vectors_added_heading"2require_relative "../science/motion_in_a_plane/vectors"34solve :two_unit_vectors_added_heading, using: Vectors do5  deduce_from "unit vectors", A: 1, B: 16  deduce_from "along x- and y- axis", phi: "90 deg"7  take direction: :beta8  answer find_value(:direction, in: "deg")910  conclusion "The two parts are equal, so their sum runs along the diagonal of a square, halfway between the " \11             "two axes."12end

playground.rb

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

The laws it drew on

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