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Find a unit vector in $\mathbb{R}^3$ that has first component 4/5. Can we describe all such vectors?

TRUSKI
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letmetalk
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    Do you know what a unit vector is? Do you know at least one example of a unit vector that is not a column of the identity matrix? Then this should be easy. – P Vanchinathan Sep 15 '17 at 05:13

3 Answers3

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Let the unit vector be $\begin{pmatrix}\frac{4}{5}\\y\\z\\ \end{pmatrix}$.
As the magnitude of an unit vector is $1$, we get $$\frac{16}{25} + y^2 + z^2 = 1 \implies y^2 + z^2 = \left ( \frac{3}{5}\right ) ^ {2}$$ Hence all the required vectors are of the form $\begin{pmatrix}\frac{4}{5}\\y\\z\\ \end{pmatrix}$ where $(y, z)$ lie on a circle of radius $\frac{3}{5}$

Shraddheya Shendre
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suppose ($4/5$ $x$ $y$) is all unit vectors starting with $4/5$ for choises of $x$ and $y$. Then solve the equation $16/25+x^2+y^2=1$ to get the value of x and y. These will be the full collection.

TRUSKI
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It is the intersection of two elements:

  • the unit sphere
  • the plane $x = \frac{4}{5}$

This should be a circle parallel to plane $yOz$ and has a radius $$ r = \sqrt{1-x^2} = \frac{3}{5} $$ The center of the circle is $(x, 0, 0)$

doraemon
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