Trong mặt phẳng tọa độ Oxy, cho hai vec tơ u = (4;1), v= (1;4). Tìm m để

* Hướng dẫn giải

Đáp án C

Ta có: $\left\{\begin{array}{c}\overrightarrow{a}=m.\overrightarrow{u}+\overrightarrow{v}=\left(4m+1;m+4\right)\\ \overrightarrow{b}=\overrightarrow{i}+\overrightarrow{j}=\left(1;1\right)\end{array}\right.$

Yêu cầu bài toán $\iff \cos \left(\overrightarrow{a},\overrightarrow{b}\right)=\cos {45}^0=\frac{\sqrt{2}}{2}$

$$\begin{gathered} \iff \frac{\left(4m+1\right)+\left(m+4\right)}{\sqrt{2}\sqrt{{\left(4m+1\right)}^2+{\left(m+4\right)}^2}}=\frac{\sqrt{2}}{2} \\ \iff \frac{5\left(m+1\right)}{\sqrt{2}\sqrt{17m^2+16m+17}}=\frac{\sqrt{2}}{2} \\ \iff 5\left(m+1\right)=\sqrt{17m^2+16m+17} \\ \iff \left\{\begin{array}{c}m+1\ge 0\\ 25m^2+50m+25=17m^2+16m+17\end{array}\right. \\ \iff m=-\frac{1}{4} \end{gathered}$$