Properties of vectors

Show that for vectors u, v and w \[ \begin{align} (u + v) + w &= u + (v + w) \end{align} \ \] and \[ \begin{align} \parallel cv \parallel &= |c| \parallel v \parallel \end{align} \ \]

Use components. For the first property

Let the components be \[ \langle u_1,u_2 \rangle, \quad \langle v_1,v_2 \rangle, \quad \langle w_1,w_2 \rangle \quad \ \] Then \[ \begin{align} (u + v) + w &=(\langle u_1,u_2 \rangle + \langle v_1,v_2 \rangle) + \langle w_1,w_2 \rangle \\ &= \langle (u_1 + v_1)+ w_1, (u_2+ v_2)+ w_2 \rangle \\ &= \langle u_1 + (v_1+ w_1), u_2+ (v_2+ w_2) \rangle \\ &= \langle u_1,u_2 \rangle + (\langle v_1,v_2 \rangle) + \langle w_1,w_2 \rangle) \\ &= u + (v+w) \end{align} \ \]

Squares are always positive

\[ \begin{align} \parallel cv \parallel &= \parallel \langle cv_1, cv_2 \rangle \parallel \\ &= \sqrt{(cv_1)^2+(cv_2)^2} \\ &= \sqrt{c^2(v_1)^2+c^2(v_2)^2} \\ &= c\sqrt{(v_1)^2+(v_2)^2} \\ &= |c|\parallel v \parallel \end{align} \ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

y

Now You Try:

Show that for vectors u and v \[ \begin{align} v + -v &= 0 \end{align} \ \] and \[ \begin{align} c(u+v) &= cu + cv \end{align} \ \]