Find the angle between the vectors for u and v given below. $$\mathbf { u } = \langle 1,1 \} , \mathbf { v } = \langle 3 , - 1 \}$$
A) $63.43 ^ { \circ }$
B) $24.09 ^ { \circ }$
C) $26.57 ^ { \circ }$
D) $69.62 ^ { \circ }$
E) $84.26 ^ { \circ }$

Question

Quizplus · Accepted Answer

The angle between two vectors $\mathbf{u}$ and $\mathbf{v}$ can be found using the dot product formula: $\cos(\theta) = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}$. For $\mathbf{u} = \langle 1,1 \rangle$ and $\mathbf{v} = \langle 3,-1 \rangle$, the dot product $\mathbf{u} \cdot \mathbf{v} = (1)(3) + (1)(-1) = 2$. The magnitudes are $\|\mathbf{u}\| = \sqrt{1^2 + 1^2} = \sqrt{2}$ and $\|\mathbf{v}\| = \sqrt{3^2 + (-1)^2} = \sqrt{10}$. Thus, $\cos(\theta) = \frac{2}{\sqrt{2}\sqrt{10}} = \frac{2}{\sqrt{20}} = \frac{1}{\sqrt{5}}$. Solving for $\theta$ gives $\theta \approx 63.43^\circ$.

Find the Angle Between the Vectors for U and V u=⟨1,1},v=⟨3,−1}\mathbf { u } = \langle 1,1 \} , \mathbf { v } = \langle 3 , - 1 \}u=⟨1,1},v=⟨3,−1}

Find the Angle Between the Vectors for U and V $\mathbf { u } = \langle 1,1 \} , \mathbf { v } = \langle 3 , - 1 \}$