Find the angle between the vectors u and v if $\mathbf { u } = \mathbf { i } + 2 \mathbf { j }$ and $\mathbf { v } = 4 \mathbf { i } + 2 \mathbf { j }$ Round answer to two decimal places.
A)36.87°
B)37.83°
C)38.28°
D)34.94°
E)35.63°

Question

Quizplus · Accepted Answer

The angle $\theta$ between two vectors $\mathbf{u}$ and $\mathbf{v}$ can be found using the dot product formula: $\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos(\theta)$, where $|\mathbf{u}|$ and $|\mathbf{v}|$ are the magnitudes of $\mathbf{u}$ and $\mathbf{v}$, respectively. For $\mathbf{u} = \mathbf{i} + 2\mathbf{j}$ and $\mathbf{v} = 4\mathbf{i} + 2\mathbf{j}$, the dot product $\mathbf{u} \cdot \mathbf{v} = (1)(4) + (2)(2) = 4 + 4 = 8$. The magnitudes are $|\mathbf{u}| = \sqrt{1^2 + 2^2} = \sqrt{5}$ and $|\mathbf{v}| = \sqrt{4^2 + 2^2} = \sqrt{20}$. Thus, $\cos(\theta) = \frac{8}{\sqrt{5}\sqrt{20}} = \frac{8}{\sqrt{100}} = \frac{8}{10} = 0.8$. Solving for $\theta$ gives $\theta = \cos^{-1}(0.8) \approx 36.87^\circ$.

Find the Angle Between the Vectors U and V If u=i+2j\mathbf { u } = \mathbf { i } + 2 \mathbf { j }u=i+2j

Find the Angle Between the Vectors U and V If $\mathbf { u } = \mathbf { i } + 2 \mathbf { j }$