Question

How do you add vectors? Add the vector <3,4> to the vector that goes 7 units at an angle of 2π/3.

Answer 1

Answer:

The sum of the two vectors is the vector <-0.5 , 4+3.5$\sqrt{3}$ >

Step-by-step explanation:

The horizontal component (x) of a vector whose magnitude is b units and its direction is Ф° is b cos Ф

The vertical component (y) of a vector whose magnitude is b and its direction is Ф is b sin Ф

The vector is <b cos Ф , b sin Ф>

∵ The vector goes 7 units at an angle $\frac{2\pi }{3}$

- That means its magnitude is 7 and its direction is  $\frac{2\pi }{3}$

∴ x = 7 cos(  $\frac{2\pi }{3}$ )

∴ y = 7 sin(  $\frac{2\pi }{3}$ )

∵ cos(  $\frac{2\pi }{3}$ ) =  $-\frac{1}{2}$

∵ sin(  $\frac{2\pi }{3}$ ) =  $\frac{\sqrt{3}}{2}$

- Substitute them in x and y

∴ x = (7)( $-\frac{1}{2}$ )

∴ x = -3.5

∴ y = (7)( $\frac{\sqrt{3}}{2}$ )

∴ y = 3.5$\sqrt{3}$

∴ The vector is <-3.5 , 3.5$\sqrt{3}$>

Now lets add the vectors by adding xs and ys components

∵ <3 , 4> + <-3.5 , 3.5$\sqrt{3}$ > = <3 + -3.5 , 4 + 3.5$\sqrt{3}$ >

∴  <3 , 4> + <-3.5 , 3.5$\sqrt{3}$> = <-0.5 , 4+3.5$\sqrt{3}$ >

∴ The sum of the two vectors is the vector <-0.5 , 4+3.5$\sqrt{3}$ >