Question

Sabendo que |u|=2, |v|=3 e u.v =-1 calcular (2v-u).(2v)

Answer 1

$\|\overrightarrow{\mathbf{u}}\|=2\,,~~\|\overrightarrow{\mathbf{v}}\|=3\,,~~\text{ e }~~\overrightarrow{\mathbf{u}}\cdot \overrightarrow{\mathbf{v}}=-1.$

Queremos calcular o seguinte produto escalar, utilizando as propriedades operatórias:

$(2\overrightarrow{\mathbf{v}}-\overrightarrow{\mathbf{u}})\cdot (2\overrightarrow{\mathbf{v}})\\ \\ =(2\overrightarrow{\mathbf{v}})\cdot (2\overrightarrow{\mathbf{v}})-\overrightarrow{\mathbf{u}}\cdot 2\overrightarrow{\mathbf{v}}\\ \\ =(2\cdot 2)\overrightarrow{\mathbf{v}}\cdot \overrightarrow{\mathbf{v}}-2\overrightarrow{\mathbf{u}}\cdot \overrightarrow{\mathbf{v}}\\ \\ =4\|\overrightarrow{\mathbf{v}}\|^{2}-2\cdot (-1)\\ \\ =4\cdot 3^{2}-2\cdot (-1)\\ \\ =4\cdot 9+2\\ \\ =36+2\\ \\ =38$