Question

Find the angle (in degrees) between the vectors. (Round your answer to two decimal places.)

Answer 1

You have to use the following formula to calculate the angle between TWO vectors:

`\cos \left(\theta \right)\:=\frac{\vec{a\:}\cdot \vec{b\:}}{\left|\vec{a\:}\right|\cdot \left|\vec{b\:}\right|}`

In this case the vector u will be the vector a, and the vector v will be the vector b

To replace the formula, we have to know the dot product:

In this case a * b

Multiply each i and each j

`\vec{a\:}\cdot\vec{b\:}=\text{ \lparen}ai*bi)+(aj*bj)`
`\vec{a\:}\cdot\vec{b\:}=\text{ \lparen}3*-7)+(4*5)=-21+20=-1`

Now

`|a|=√(ai^2+aj^2)=√(3^2+4^2)=√(25)=5`
`|b|=√(bi^2+bj^2)=√((-7)^2+5^2)=√(74)`

Now replace in

`\cos(\theta)=\frac{\vec{a}\vec{b}}{\lvert\vec{a}\rvert\lvert\vec{b}\rvert}`
`\cos \left(θ\right)=-(1)/(5√(74))`

Clear Cos with ArcCos

`θ=\arccos\left(\cos\left(θ\right)\right)=\arccos\left(-(1)/(5√(74))\right)`

ANS:

`θ=\text{ 91.33221985\degree}`