If $ \ S : \mathbb{R}^3 \to \mathbb{R}^3 \ $ be a shear map with respect to the unit vector $ \ n \ $

Question

Select the correct options :

(1) If $ \ S : \mathbb{R}^3 \to \mathbb{R}^3 \ $ be a shear map with respect to the unit vector $ \ n \ $ , then $ \ S (x)=x \ $ for every $ \ x \perp n \ $

(2) If $ \ S : \mathbb{R}^3 \to \mathbb{R}^3 \ $ be a shear map with respect to the unit vector $ \ n \ $ , then $ \ || S (x)|| =||x|| \ $ for every $ \ x \perp n \ $

(3) If $ \ S : \mathbb{R}^3 \to \mathbb{R}^3 \ $ be a shear map with respect to the unit vector $ \ n \ $ , then $ \ S (x)=x \ $ for every $ \ x \in \mathbb{R}^3 \ $

Answer:

We know that a shear map shift a vector along a particular direction.

But I can not answer the above questions.

Can some one help me with atleast hints?

Answer 1

HINT

In the basis $\{m_1,n,m_2\}$ with $m_1\perp n$ and $m_2\perp n,m_1$ we have that

$$T_S=\begin{bmatrix}1&k&0\\0&1&0\\0&0&1\end{bmatrix}$$