Question

Assume that ||v|| = 2 and ||w|| = 3, and the angle between v and w is 120 degrees. Determine:(a) v·w
(b) ∥2v+w∥
(c) ∥2v−3w∥

Answer 1

Answer:

Step-by-step explanation:

Given that  ||v|| = 2 and ||w|| = 3, and the angle between v and w is 120 degrees

We know by definition of dot product and properties that

$a.b = ||a||||b||cos t$ where t is the angle between them

Using this we find

(a) v·w  =$2(3) cos 120\\=-3$

(b) ∥2v+w∥

$(2v+w).(2v+w) = 4 v.v +4 w.v +w.w= 4(2)^2+4(-3)+9 =13$

$||2v+w||^2 =13\\||2v+w||=\sqrt{13}$

(c) ∥2v−3w∥

$(2v-3w).(2v-3w) = 4 v.v -12 w.v +9w.w= 4(2)^2-12(-3)+9(9) =133$

$||2v-3w||^2 =133\\||2v-3w||=\sqrt{133}$