Question

Vectors u = −4(cos 60°i + sin60°j), v = 8(cos 135°i + sin135°j), and w = 12(cos 150°i + sin150°j) are given. Use exact values when evaluating sine and cosine.Part A: Find −7u • v. Show your work. (4 points)Part B: Use the dot product to determine if u and w are parallel, orthogonal, or neither. Justify your answer. (6 points)

Answer 1

Given the following vectors:

`\begin{gathered} u=-4\mleft(\cos 60\degree i+\sin 60\degree j\mright) \\ v=8\mleft(\cos 135\degree i+\sin 135\degree j\mright) \\ w=12\mleft(\cos 150\degree i+\sin 150\degree j\mright) \end{gathered}`

We will find the following:

a) -7u • v

The dot product of two vectors will be as follows:

`A\cdot B=A_xB_x+A_yB_y`

So, for the given product, the answer will be as follows:

`\begin{gathered} -7u\cdot v=-7(-4)(8)(\cos 60\cdot\cos 135+\sin 60\cdot\sin 135) \\ =224((1)/(2)\cdot(-\frac{1}{\sqrt[]{2}})+\frac{\sqrt[]{3}}{2}\cdot\frac{1}{\sqrt[]{2}}) \\ \\ =224(-\frac{1}{2\sqrt[]{2}}+\frac{\sqrt[]{3}}{2\sqrt[]{2}})\cdot\frac{\sqrt[]{2}}{\sqrt[]{2}} \\ \\ =224(-\frac{\sqrt[]{2}}{4}+\frac{\sqrt[]{6}}{4})=56(\sqrt[]{6}-\sqrt[]{2}) \end{gathered}`

Part B: Use the dot product to determine if u and w are parallel, orthogonal, or neither

The vectors will be parallel if the dot product = the product of the magnitudes which means the angle between the vectors = 0 or 180

And the vectors will be orthogonal of the dot product = 0

This means the angle between them = 90

The dot product of the vectors (u) and (w) will be as follows:

`\begin{gathered} u\cdot w=(-4)(12)(\cos 60\cos 150+\sin 60\sin 150) \\ =(-48)((1)/(2)\cdot\frac{-\sqrt[]{3}}{2}+\frac{\sqrt[]{3}}{2}\cdot(1)/(2)) \\ \\ =(-48)(\frac{-\sqrt[]{3}}{4}+\frac{\sqrt[]{3}}{4})=(-48)(0)=0 \end{gathered}`

So, as the result of the dot product = 0

The vectors (u) and (w) are Orthogonal.