IDNLearn.com is your go-to resource for finding answers to any question you have. Our experts provide accurate and detailed responses to help you navigate any topic or issue with confidence.
Answer:
Approximately [tex]2.3127[/tex] radians, which is approximately [tex]132.51^{\circ}[/tex].
Step-by-step explanation:
Dot product between the two vectors:
[tex]\begin{aligned}& u\cdot v \\ =\; & 7 \times (-1) + (-2) \times 2 \\ =\; & (-11) \end{aligned}[/tex].
Magnitude of the two vectors:
[tex]\begin{aligned} \| u \| &= \sqrt{{7}^{2} + {(-2)}^{2}} \\ &= \sqrt{53} \end{aligned}[/tex].
[tex]\begin{aligned} \| v \| &= \sqrt{{(-1)}^{2} + {2}^{2}} \\ &= \sqrt{5} \end{aligned}[/tex].
Let [tex]\theta[/tex] denote the angle between these two vectors. By the property of dot products:
[tex]\begin{aligned} \cos(\theta) &= \frac{u \cdot v}{\|u\| \, \| v \|} \\ &= \frac{(-11)}{(\sqrt{53})\, (\sqrt{5})} \\ &= \frac{(-11)}{\sqrt{265}}\end{aligned}[/tex].
Apply the inverse cosine function [tex]{\rm arccos}[/tex] to find the value of this angle:
[tex]\begin{aligned} \theta &= \arccos\left(\frac{u \cdot v}{\| u \| \, \| v \|}\right) \\ &= \arccos\left(\frac{(-11)}{\sqrt{265}}\right) \\ & \approx \text{$2.3127$ radians} \\ &= 2.3127 \times \frac{180^{\circ}}{\pi} \\ &\approx 132.51^{\circ}\end{aligned}[/tex].