Question

given ( u=(3,2), v=(1,5) ), and the angle between the vectors is ( 45^{circ} ), what is ( ucdot v )?
3
4
13
21

Explanation:

Step1: Recall the formula for the dot product

The dot product of two vectors \(\mathbf{u}=(u_1,u_2)\) and \(\mathbf{v}=(v_1,v_2)\) is given by \(\mathbf{u}\cdot\mathbf{v}=u_1v_1 + u_2v_2\).

Step2: Substitute the components of the vectors

For \(\mathbf{u}=(3,2)\) and \(\mathbf{v}=(1,5)\), we have \(u_1 = 3\), \(u_2=2\), \(v_1 = 1\), \(v_2 = 5\).
Then \(\mathbf{u}\cdot\mathbf{v}=(3\times1)+(2\times5)\).

Step3: Calculate the result

\(3\times1=3\) and \(2\times5 = 10\).
So \(\mathbf{u}\cdot\mathbf{v}=3 + 10=13\).

(The information about the angle \(45^{\circ}\) is a distractor in this problem as we are directly using the formula for the dot - product of two vectors in component form \(\mathbf{u}\cdot\mathbf{v}=u_1v_1+u_2v_2\) and not using the formula \(\mathbf{u}\cdot\mathbf{v}=\vert\mathbf{u}\vert\vert\mathbf{v}\vert\cos\theta\))

Answer:

13