HW 3.3 – Dot Product - solutions

Find the dot product1. (5,-1)·(2,4)

= (5)(2)+(-1)(4) = 6

2. (−27

−4)⋅(20

−3)= (-2)(2)+(7)(0)+(-4)(-3) = 8

Find the angle between the two vectors [in radians!]3. (6,2) & (4,-1)

cos(ϑ) =(6,2)⋅(4,−1)

∣(6,2)∣⋅∣(4,−1)∣=

22√ 40⋅√17

=22

√680≈ .844 so ϑ ≈ .567

4. (2,7) & (-4,-14)one vector is a multiple of the other (in the opposite direction) so ϑ = π

5. (4,-3,5) & (-2,-1,1)

cos(ϑ) =(4,−3,5)⋅(−2,−1,1)

∣(4,−3,5)∣⋅∣(−2,−1,1)∣=

0√a⋅√b

=0 so ϑ = π/2

6. (2,-3,6) & (-1,5,2)

cos(ϑ) =(2,−3,6)⋅(−1,5,2)

∣(2,−3,6)∣⋅∣(−1,5,2)∣=

−5√49⋅√30

=−57√30

≈−.130 so ϑ ≈ 1.702

Other problems7. find the component of (3,4) in the (a) x-direction (b) y-direction (c) direction of (4,-2)

(a) (3,0) or 3 i⃗ (b) 4 j⃗ (c)

8. find the component of (2,-3,5) in the (a) x-direction (b) y-direction (c) z-direction (d) direction of (4,3,-2)

a) 2 i⃗ b) −3 j⃗ c) 5 k⃗

d)

9. Suppose a truck is towing a car. The truck exerts a force of 5000N, and the tow chain makes an angle of 35o with the ground. (a) Find the magnitude of your pulling force which is doing work. (b) Find the work done to tow the car 50 meters.

(a) Magnitude of Pulling force = 5000 cos(35o) = 4095.76 N(b) work W = |F|cos(t)·|D| = 4095.76 · 50 = 204788 J

10. If you pull a table with a force vector (in Newtons) of (15,8) and the table moves with a displacement vector (in meters) of (3,5), find the work done.

W = F⃗⋅D⃗ = (15,8) · (3,5) = 85 J