Mylo O'Moore

2021-02-06

A vector is first rotated by $90}^{\circ$ along x-axis and then scaled up by 5 times is equal to $(15,-10,20)$ . What was the original vector

lobeflepnoumni

Skilled2021-02-07Added 99 answers

Let us consider the vector (x, y, z).

The vector is first rotated by$90}^{\circ$ along x - axis and then scaled up by 5 times.

Ratation: In homogeneous coordinates, the matrix for a rotetion about the origin through a given angle$\theta \text{}is\text{}B=\left[\begin{array}{ccc}\mathrm{cos}\theta & -\mathrm{sin}\theta & 0\\ \mathrm{sin}\theta & \mathrm{cos}\theta & 0\\ 0& 0& 1\end{array}\right]$

The vector (x, y, z) is rotated by$90}^{\circ$ along x-axis then the vector becomes

$\left[\begin{array}{c}\mathrm{cos}{90}^{\circ}-\mathrm{sin}{90}^{\circ}0\\ \mathrm{sin}{90}^{\circ}\mathrm{cos}{90}^{\circ}\\ 001\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

$=\left[\begin{array}{c}0-10\\ 100\\ 001\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

$=\left[\begin{array}{c}-y\\ x\\ z\end{array}\right]$

Now, the vector is scaled up by 5 times.

Scaling: In homogeneous coordinates, a scaling about the origin with scale factors c in the x-direction and c in the y-direction is computed by multiplying the vector by the matrix

$A=\left[\begin{array}{c}c00\\ 0c0\\ 001\end{array}\right]$

Thus, the vector becomes

$\left[\begin{array}{c}500\\ 050\\ 001\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

$=\left[\begin{array}{c}-5y\\ 5x\\ z\end{array}\right]$

Now, given that the reduced vector is$(15,-10,20)$ . Therefore, we have

$\left[\begin{array}{c}-5y\\ 5x\\ z\end{array}\right]=\left[\begin{array}{c}15\\ -10\\ 20\end{array}\right]$

By the equality of two matrices we have

$\{\begin{array}{c}-5y=15\\ 5x=-10\\ z=20\end{array}=\{\begin{array}{c}y=-3\\ x=-2\\ z=20\end{array}$

Thus, the original vector is (-2, -3, 20)

The vector is first rotated by

Ratation: In homogeneous coordinates, the matrix for a rotetion about the origin through a given angle

The vector (x, y, z) is rotated by

Now, the vector is scaled up by 5 times.

Scaling: In homogeneous coordinates, a scaling about the origin with scale factors c in the x-direction and c in the y-direction is computed by multiplying the vector by the matrix

Thus, the vector becomes

Now, given that the reduced vector is

By the equality of two matrices we have

Thus, the original vector is (-2, -3, 20)

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