Identify logical statements (MATH)

Geometry Level 3

Which statement is geometrically false?

A. If $a // (\alpha)$ and $b \perp a$ then $b \perp (\alpha)$
B. If $a // b$ and $(\alpha) \perp a$ then $(\alpha) \perp b$
C. If $(\alpha) // (\beta)$ and $a \perp (\alpha)$ then $a \perp (\beta)$
D . If $a \perp (\alpha)$ and $b \perp (\alpha)$ , then $a // b$ .

Note: $(\beta), (\alpha)$ denote planes. $a,b$ denote lines. Objects with different names are distinct from each other.

Bonus: Explain why.

A C All statements are true. All statements are false. B D

1 solution

Richard Desper
Feb 25, 2020

A: let $a = \{ (s,0,1): s \in \Re\}$ , and $\alpha$ be the $x-y$ plane $\{(x,y,0): x, y \in \Re\}$ .
If $b = \{(0,t,1): t \in \Re\}$ , then $a \perp b$ , but $a \parallel \alpha$ and $b \parallel \alpha$ .

The space of lines perpendicular to $a$ contains only one line perpendicular to $\alpha$ , and infinitely many lines that are not.

The other three statements are true.

Identify logical statements (MATH)

1 solution

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