Question

Find the angle between the vectors u = 4i – 3j and v = 8i + 2j.

Answer 1

Given:

Two vectors u = 4i – 3j and v = 8i + 2j.

Required:

To find the angle between the given vectors.

Step-by-step explanation:

The dot product between two vectors is given by the formula:

`a.b=\lvert{a}\rvert\lvert{b}\rvert cos\theta`

Now first we will find the dot product of the given vectors.

`\begin{gathered} u.v=(4i-3j).(8i+2j) \\ u.v=(4*8)-(3*2) \\ u.v=32-6 \\ u.v=26 \end{gathered}`
`\begin{gathered} \lvert{u}\rvert=√((4)^2+(-3)^2) \\ \lvert{u}\rvert=√(16+9) \\ \lvert{u}\rvert=√(25) \\ \lvert{u}\rvert=5 \end{gathered}`
`\begin{gathered} \lvert{v}\rvert{=√((8)^2+(2)^2)}\rvert \\ \lvert{v}\rvert=√(64+4) \\ \lvert{v}\rvert=√(68) \end{gathered}`

Now put these values in the dot product formula:

`\begin{gathered} 26=5*√(65)* cos\theta \\ cos\theta=(26)/(5√(68)) \\ \theta=cos^(-1)((26)/(5√(68))) \end{gathered}`

Final Answer:

Thus the angle between the given vectors is:

`\theta=cos^(-1)((26)/(5√(68)))`