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
here tension in the string is counter balanced by weight of block of mass m1
so we can say


now on the other side the block which is placed on the inclined plane
we can say that component of weight of the block and friction force is counter balanced by tension force

now we can plug in all values to find the friction force


so it will have 0.83 N force on it due to friction
now to find the friction coefficient

here we know that


now from above equation


so friction coefficient will be 0.38
Answer:
the thermal energy generated in the loop = 
Explanation:
Given that;
The length of the copper wire L = 0.614 m
Radius of the loop r = 
r = 
r = 0.0977 m
However , the area of the loop is :



Change in the magnetic field is 
Then the induced emf e = 
e = 
e = 2.74 × 10⁻³ V
resistivity of the copper wire
Ω m
diameter of the wire = 1.08 mm
radius of the wire = 0.54 mm = 0.54 × 10⁻³ m
Thus, the resistance of the wire R = 
R = 
R = 1.13× 10⁻² Ω
Finally, the thermal energy generated in the loop (i.e the power) = 
= 
= 
Answer:

The Required horizontal force is 230.04N
Explanation:
Since the velocity is constant so acceleration is zero; a=0
Now the horizontal force required to move the pickup is equal to the frictional force.

where:
F_{Hn} is the required Force
u is the friction coefficient
m is the mass
g is gravitational acceleration=9.8m/s^2
Eq (1)
Now, weight increases by 42% and friction coefficient decreases by 19%
New weight=(1.42*m*g) and new friction coefficient=0.81u
Eq (2)
Divide Eq(2) and Eq (1)

The Required horizontal force is 230.04N