Find the dot product of $\vec { u } = \langle 4,7 \rangle$ and $\vec { v } = \langle 1,3 \rangle$ and the cosine of the angle between them.
A) $10 , \cos ( \theta ) = \frac { 1 } { 13 }$
B) $25 , \cos ( \theta ) = \frac { 5 } { \sqrt { 26 } }$
C) $25 , \cos ( \theta ) = \frac { 1 } { 25 }$
D) $5 , \cos ( \theta ) = \frac { 5 } { \sqrt { 26 } }$

Question

Quizplus · Accepted Answer

The dot product of $\vec{u}$ and $\vec{v}$ is calculated as $4 \times 1 + 7 \times 3 = 4 + 21 = 25$. The cosine of the angle between them is given by $\cos(\theta) = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\| \|\vec{v}\|}$, where $\|\vec{u}\| = \sqrt{4^2 + 7^2} = \sqrt{16 + 49} = \sqrt{65}$ and $\|\vec{v}\| = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}$. Thus, $\cos(\theta) = \frac{25}{\sqrt{65} \sqrt{10}} = \frac{25}{\sqrt{650}} = \frac{25}{\sqrt{26} \cdot 5} = \frac{5}{\sqrt{26}}$.

Find the Dot Product Of u⃗=⟨4,7⟩\vec { u } = \langle 4,7 \rangleu=⟨4,7⟩

Find the Dot Product Of $\vec { u } = \langle 4,7 \rangle$