Vector addition and scalar multiplication

The two basic vector operations — combine two vectors componentwise, or stretch/shrink/reverse one by a number.

Addition and scalar multiplication are both defined componentwise: $\mathbf{u}+\mathbf{v} = \langle u_1+v_1,\,u_2+v_2,\,u_3+v_3\rangle$ and $k\mathbf{v}=\langle kv_1,\,kv_2,\,kv_3\rangle$ for a scalar $k$ . The sign of $k$ controls direction, not just size: $k>0$ keeps $\mathbf{v}$ 's direction, $k<0$ reverses it exactly $180°$ , and $k=0$ collapses it to the zero vector, the only vector with no direction at all.

Examples

For $\mathbf{u}=\langle 1,-3,2\rangle$ , $\mathbf{v}=\langle -4,0,1\rangle$ , $2\mathbf{u}-3\mathbf{v}=\langle 14,-6,1\rangle$ .