Question

Use the alternative form of the dot product to find u · v.

||u||=45 , ||v||=30 and the angle between u and v is 5π/6.

(b) Find the angle θ between the vectors in radians and in degrees.
u = cos( pi/3) i + sin(pi/3) J , v=Cos (3pi/4)I + sin (3pi/4) J , find radians and degrees with respect to theta

c.) Find the direction cosines and angles of u. Notice that the sum of the squares of the direction cosines is 1. (Round your answers for the angles to four decimal places.)

u = i + 4j + 8k

cos α, cos β, and cos γ,

Answer 1

Explanation:

u = 45

v = 30

Angle between them, θ = 5π / 6 = 150°

(a) The formula for the dot product is given by

`\overrightarrow{u}.\overrightarrow{v}= u v Cos\theta`

`\overrightarrow{u}.\overrightarrow{v}= 45 * 30 * Cos150`

`\overrightarrow{u}.\overrightarrow{v}= -1169.13`

(b)
`\overrightarrow{u}=Cos(\pi )/(3)\widehat{i}+Sin(\pi )/(3)\widehat{j}`

`\overrightarrow{u}=0.5\widehat{i}+0.866\widehat{j}`

`\overrightarrow{v}=Cos(2\pi )/(3)\widehat{i}+Sin(2\pi )/(3)\widehat{j}`

`\overrightarrow{v}=-0.707\widehat{i}+0.707\widehat{j}`

Let the angle between them is θ

The formula in terms of the dot product is given by

`Cos\theta =\frac{\overrightarrow{u}.\overrightarrow{v}}{u v}`

`u=\sqrt{0.5^(2)+0.866^(2)}=1`

`v=\sqrt{-0.707^(2)+0.707^(2)}=1`

`Cos\theta =(0.5 * (-0.707) + 0.866 * 0.707))/(1* 1)`

`Cos\theta=0.2587`

θ = 75°

(c)
`\overrightarrow{u}=1\widehat{i}+4\widehat{j}+8\widehat{k}`

`u=\sqrt{1^(2)+4^(2)+8^(2)}=9`

`Cos\alpha =(1)/(9)`

`Cos\beta =(4)/(9)`

`Cos\gamma  =(8)/(9)`