Get the solution to "Below are various vectors in cartesian, cylindrical and..."

College
Algebra
Linear algebra
Alternate coordinate systems

jernplate8

Answered question

2021-08-02

Below are various vectors in cartesian, cylindrical and spherical coordinates. Express the given vectors in two other coordinate systems outside the coordinate system in which they are expressed
$a) \vec{A} (x, y, z) = {\vec{e}}_{x}$
$d) \vec{A} (ρ, ϕ, z) = {\vec{e}}_{ρ}$
$g) \vec{A} (r, θ, ϕ) = {\vec{e}}_{θ}$
$j) \vec{A} (x, y, z) = \frac{- y {\vec{e}}_{x} + x {\vec{e}}_{y}}{x^{2} + y^{2}}$

Answer & Explanation

ottcomn

Skilled2021-08-03Added 97 answers

Step 1
Given: $a) \vec{A} (x, y, z) = {\vec{e}}_{x}$
To express it in other two coordinate systems:
ie., in terms of cartesian, cylindrical and spherical coordinates
Step 2
Given: $a) \vec{A} (x, y, z) = {\vec{e}}_{x}$
spherical coordinates can be expressed as
$(r, θ, ϕ) = (\sqrt{x^{2} + y^{2} + z^{2}}, \tan^{- 1} (\frac{y}{x}), \cos^{- 1} (\frac{z}{\sqrt{x^{2} + y^{2} + z^{2}}}))$
Therefore,
$(r, θ, ϕ) = (1, 0, \frac{π}{2})$

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