1. Consider two vectors S (2, 1, 2) and T (5, -1, 3). a) What is the angle betwe

Question

1. Consider two vectors S (2, 1, 2) and T (5, -1, 3).
a) What is the angle between the vectors S and T?
b) Express the following using unit vectors: R = 3S – 2T

2. A 4.0 kg box, initially at rest, starts sliding down from the top of an inclined ramp that makes an angle of 25° with the horizontal. The ramp is 10 m long.

a) If the coefficient of kinetic friction is 0.35, how fast will the box be moving when it reaches the bottom of the ramp?

A block of mass m is pulled at constant velocity along a flat floor by a force T which is at an angle from the flat floor. The vertical/normal force exerted on the block by the floor is equal to:
A. ( mg)    
B. (mg + T cos)    
C.(mg - Tcos)     
D. (mg - Tsin)      
E. (mg + Tsin)

Explanation / Answer

1) Given,

S(2,1,2) and T(5,-1,3)

a)in Polar form

S = 2i + 1j + 2 k ; T = 5i - 1j + 3 k

magnitudes will be:

S = sqrt (2^2 + 1^2 + 2^2) = 3

T = sqrt (5^2 + -1^2 + 3^2) = 5.92

the dot product of two vectors gives angle between them so,

cos(theta) = a.b/lal.lbl

cos(theta) = (2i + 1j + 2k).(5i - 1j + 3k)/3 x 5.92 = 15/17.76 = 0.8446

theta = cos^-1(0.8446) = 32.37 deg

Hence, theta = 32.37 deg

b) R = 3S - 2 T

R = 3(2i + 1j + 2k)- 2(5i - 1j + 3 k)

R = 6i + 3j + 6k - 10i + 2j - 6k = -4i + 5j

Hence, R = -4i + 5 j

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