1. Commutative law is valid for the cross product of two vectors. (Commutative law: PxQ = QxP; for two vectors P and Q)
a) True
b) False
Explanation: This statement is wrong. It is not possible unless we apply a negative sign to the RHS of the equation. That is PxQ = -(QxP). It is because, if you curl your wrist from one vector towards another vector, the thumb projected will give the direction of the cross product. Thus if you reverse the direction, a negative sign is necessary
2. Which among the following is the distributive law for the cross product of three vectors?
a) Px(Q+S) = (PxQ) + (PxS)
b) Px(QxS) = (PxQ) + (PxS)
c) Px(QxS) = (PxQ) x (PxS)
d) Px(Q+S) = (PxQ) + (QxS)
Explanation: The distributive law works just like the simple multiplication of the constant before the brackets. That is in the equation Px(Q+S) = (PxQ) + (PxS), P is crossed by Q and S both. This is simple as, if we first add the two vectors and then do the cross product or we first do the cross product and the do the sum
3. Which statement is true? (For three vectors P, Q and R)
a) Associative law for cross product: (PxQ)xS = Px(QxS)
b) Associative law for cross product: (PxQ)xS ≠ Px(QxS)
c) Associative law for cross product: (PxQ)xS > Px(QxS)
d) Associative law for cross product: (PxQ)xS < Px(QxS)
Explanation: The associative law is defined in the cross product of three vectors. This property is though valid for the dot product. But for the cross product, it is not true. It is because, in the dot product the final result is the scalar quantity, but in a cross product it is the direction too. Thus the answer.
4. Which of the following is true?
a) i x i =1
b) j x i = -j
c) k x j = -i
d) k x k = 1
Explanation: As the mathematic equation for the cross product is having a cosine function in it, in which the angle used in the function is the angle between the vectors. Thus the cross product will be zero if the angle between them is 90.
5. Which of them is not correct?
a) j x j = 0
b) j x k = i
c) j x i = k
d) j x i = -k
Explanation: As asked, the one which is not correct is the third one. The product is containing the cosine function, and the angle which is going to be inserted in the function is the angle between the vectors. Thus if the angle is 90, then the cross will be zero
6. The ___________ forces do not cause the rotation.
a) Non-concurrent
b) Concurrent
c) Parallel
d) Non-Parallel
Explanation: The concurrent forces are the which are somewhere touching the axis of rotation. If any of the force is touching that axis, that force is not considered or is insufficient to cause a rotation. If a force is concurrent then the perpendicular distance of the force from the line of an axis is zero, thus no rotation. As we know rotation is caused by a moment.
7. The tendency of a force to rotate the body is called the moment of the force.
a) True
b) False
Explanation: The moment of the force about an axis or a point gives the measure of the tendency of the force. It is the cause of the body’s rotation, about that point or at that axis. Thus, the tendency of a force to rotate the body is called the moment of the force.
8. The moment is the cross product of which two vectors?
a) Force and Radius vectors
b) Radius and Force vectors
c) Force and Radius scalars
d) Radius and Force scalars
Explanation: The cross product needs to take in the proper sequence. If not taken then the answer is just the opposite of the true answer. That’s why, the answer is not the Force and Radius vectors, but the Radius and Force vectors. Because the moment has its direction, as many of the cross products have, and thus precaution needs to be taken
9. What is Varigon’s Theorem? (M = Moment, F= Force vector, R= Radius vector)
a) M = R x F (F = F1 + F2 + F3 + … vectorially adding all the forces)
b) |A||B|sinØ
c) AxB)xS ≠ Px(QxS)
d) Px(Q+S) = (PxQ) + (PxS)
Explanation: For many vectors acting on the body, Varigon’s Theorem is defined as M = R x F (F = F1 + F2 + F3 + … vectorially adding all the forces). This is just the giving the cross product of individual force and radius vectors and thus the sum is the total moment produced
10. What is the mixed triple product of three vectors?
a) S.(PxQ)
b) Sx(PxQ)
c) S.(P.Q)
d) Sx(P.Q)
Explanation: It is the scalar expression. It is the dot product of the vector to the vector, which is the result of the cross vector product of two vectors. And if two vectors are held together in the equation by a dot “.”, then the answer which you get is the scalar quantity.