Is it true ?

Geometry Level 2

If you rotate a point ( x , y , z ) (x, y, z) by 18 0 ∘ 180^{\circ} about any axis, no matter what its direction is, then the image of rotation will be the point ( − x , − y , − z ) (-x, -y, -z) .

True False

Hosam Hajjir by Hosam Hajjir , 56, Canada

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2 solutions

David Vreken
Jul 2, 2019

One of many counterexamples is to rotate the point ( 1 , 0 , 0 ) (1, 0, 0) by 180 ° 180° around the x x -axis, so that its image also has the coordinates ( 1 , 0 , 0 ) (1, 0, 0) (and not ( − 1 , 0 , 0 ) (-1, 0, 0) as predicted by the statement). Therefore, the statement is false .

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Hosam Hajjir
Jul 2, 2019

This is False. The statement is true only if the point ( x , y , z ) (x, y, z) lies in a plane whose normal vector is the axis of rotation.
In other words, the statement is true only if a ⃗ ⋅ ( x , y , z ) = 0 \vec{ a } \cdot ( x, y, z) = 0 , where a ⃗ \vec{a} is the axis vector.

2 Helpful 0 Interesting 1 Brilliant 0 Confused

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