Question

Assume that a vector ~a of length ||~a|| = 3 units. In addition, ~a points in a direction that is 135◦ counterclockwise from the positive x-axis, and a vector ~b in the xy-plane has a length ||~b|| = 1/3 and points in the positive y-direction. Find ~a · ~b.

Answer 1

$$\vec a=\langle3\cos135\degree, 3\sin 135\degree\rangle,$$

$$\vec b=\langle0,\dfrac{1}{3}\rangle$$

$$\vec a\cdot \vec b=3(-\dfrac{\sqrt{2}}{2})(0)+3(\dfrac{\sqrt{2}}{2})(\dfrac{1}{3})=\dfrac{\sqrt{2}}{2}$$

Or

Angle between two vectors $\angle(\vec a, \vec b)=135\degree-90\degree =45\degree$

$$\vec a\cdot \vec b=|\vec a||\vec b|\cos(\angle(\vec a\cdot \vec b))$$

$$=3(\dfrac{1}{3})\cos 45\degree=\dfrac{\sqrt{2}}{2}$$

$\vec a\cdot \vec b=\dfrac{\sqrt{2}}{2}$