Question

True/False: If $k$ and $m$ are scalars and $\mathbf{u}$ and $\mathbf{v}$ are vectors, then $$ (k+m)(\mathbf{u}+\mathbf{v})=k \mathbf{u}+m \mathbf{v} $$

Answer 1

True. This statement follows from the distributive property of vector addition:

$$(k+m)(\mathbf{u}+\mathbf{v}) $$

$$= k(\mathbf{u}+\mathbf{v}) + m(\mathbf{u}+\mathbf{v}) $$

$$= k\mathbf{u} + k\mathbf{v} + m\mathbf{u} + m\mathbf{v} $$

$$= k\mathbf{u} + m\mathbf{v}$$

Therefore, $(k+m)(\mathbf{u}+\mathbf{v}) = k \mathbf{u}+m \mathbf{v}$.