Recall the definition of the cross product with respect to the unit vectors:
i × i = j × j = k × k = 0
i × j = k
j × k = i
k × i = j
and that the product is anticommutative, so that for any two vectors u and v, we have u × v = - (v × u). (This essentially takes care of part (b).)
Now, given a = 8i + j - 2k and b = 5i - 3j + k, we have
a × b = (8i + j - 2k) × (5i - 3j + k)
a × b = 40 (i × i) + 5 (j × i) - 10 (k × i)
… … … … - 24 (i × j) - 3 (j × j) + 6 (k × j)
… … … … + 8 (i × k) + (j × k) - 2 (k × k)
a × b = - 5 (i × j) - 10 (k × i) - 24 (i × j) - 6 (j × k) - 8 (k × i) + (j × k)
a × b = - 5k - 10j - 24k - 6i - 8j + i
a × b = -5i - 18j - 29k
Answer: Trotting uses only 75 percent of the energy as galloping
Explanation: Trotting is only 300 J/m, whereas galloping is roughly 400 J/m
Answer:
(a) 47.08°
(b) 47.50°
Explanation:
Angle of incidence = 78.9°
<u>For blue light :
</u>
Using Snell's law as:
Where,
Θ₁ is the angle of incidence
Θ₂ is the angle of refraction
n₂ is the refractive index for blue light which is 1.340
n₁ is the refractive index of air which is 1
So,
Angle of refraction for blue light = sin⁻¹ 0.7323 = 47.08°.
<u>For red light :
</u>
Using Snell's law as:
Where,
Θ₁ is the angle of incidence
Θ₂ is the angle of refraction
n₂ is the refractive index for red light which is 1.331
n₁ is the refractive index of air which is 1
So,
Angle of refraction for red light = sin⁻¹ 0.7373 = 47.50°.
Answer:
the film's minimum thickness to make the reflected light enhanced in brightness is 112.8 nm
Explanation:
Given the data in the question;
wavelength λ = 600 nm
index of refraction = 1.33
so the film's minimum thickness to make the reflected light enhanced in brightness = ?
condition for maxima for reflected light is;
2nt = [ ( 2m + 1 )λ ] / 2
we solve for t
4nt = [ ( 2m + 1 )λ ]
t = [ ( 2m + 1 )λ ] / nt
Now, for minimum thickness m = 1
t = λ / 4n
so we substitute in our values;
t = 600 nm / ( 4 × 1.33 )
t = 600 nm / 5.32
t = 112.78195 ≈ 112.8 nm { one decimal place }
Therefore, the film's minimum thickness to make the reflected light enhanced in brightness is 112.8 nm