# Theory of Machines Questions and Answers – Position of a Point

This set of Theory of Machines Multiple Choice Questions & Answers (MCQs) focuses on “Position of a Point”.

1. If the path of a moving point can be described by only two coordinates, then the point is said to have ______
a) Rectilinear motion
b) Spatial motion
c) Skew motion
d) Planar motion

Explanation: If the coordinate axes are chosen such that one coordinate is always zero or constant, then the path is contained in a single plane and the point is said to have planar motion. Space and skew motion is usually the same, they require three coordinates. The rectilinear motion requires only a single coordinate, i.e. two of its spatial coordinates are zero or constant and the point moves in a straight line.

2. Define the position of a point.
a) The vector from the origin of a specified reference coordinate system to the point
b) The distance of the point from the origin of a specified reference system
c) Angular orientation made by the point with any one of the planes of the reference coordinate system
d) The vector from the point to the origin of a specified reference coordinate system.

Explanation: Vector has direction. Hence, it is important to note that the position of a point is a vector from the origin of the reference coordinate system to the point. In order to define the position, we require two properties, both magnitude, and direction, which are precisely required for a vector.

3. Position of a point is a relative concept.
a) False
b) True

Explanation: Position of a point is always determined using magnitude and direction of a point relative to the reference coordinate system. Hence, it is a relative concept.

4. If Rx, Ry, Rz be the distance of point R from x, y, and z-axes respectively and i, j and k be the unit vectors in that direction then position of point R is given by ______
a) R = Rx i + Ry j + Rz k
b) R = Rx j + Ry i + Rz k
c) R = Rx k + Ry j + Rz i
d) R = Rx i + Ry k + Rz j

Explanation: Position of a point is given by the combination of both magnitude and direction, in which the direction is justified with the unit vector in that direction.

5. If Rx, Ry, Rz be the distance of point R from x, y and z-axes respectively then magnitude of the position of point R is not given by which of the following ______
a) |R|
b) √(R.R)
c) Rx i + Ry j + Rz k
d) √(Rx2 + Ry2 + Rz2)

Explanation: Rx i + Ry j + Rz k is the position vector of the point R. Other options are used to derive the magnitude of distance between the point R and origin of the reference coordinate system.
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6. Unit vector is given by ______
a) Any vector whose magnitude is unity
b) The ratio of the magnitude of the vector to the position vector
c) None of these
d) The ratio of position vector to the magnitude of that vector

Explanation: Unit vector is a vector whose magnitude should be equal to unity and should only possess the direction of a certain vector. Mathematically, it is achieved by dividing the given vector by its own magnitude so that only its direction exists.

7. Find the unit vector of i -2j + 2k.
a) 3i – 6j + 6k
b) $$\frac{1}{3}i-\frac{2}{3}j+\frac{2}{3}k$$
c) $$\frac{2}{3}i+\frac{1}{3}j-\frac{2}{3}k$$
d) 6i + 3j – 6k

Explanation: Let A = i – 2j + 2k
Magnitude of A = |A| = $$\sqrt{1^2+(-2)^2+2^2}$$
= √9 = 3
Unit vector of A = $$\frac{i – 2j + 2k}{3}$$

8. Two vectors are given by: Rp = i + 2j + k and RQ = 3i + 4j + k. Position vector RQ with respect to RP is given by ______
a) 2(i + j)
b) 2(i + k)
c) i + j
d) 2(j + k)

Explanation: The position difference between two points is given by vector difference of one vector with respect to another vector.
RQP = RQ – RP = (3-1) i + (4-2) j + (1-1) k = 2i + 2j

9. The path of moving point is defined by the equation y=4x2. Find the position difference from point P to point Q. If RPX = 4 and RQX = 5.
a) i + 36j
b) 9i + 164j
c) – (9i – 164j)
d) – (i + 36j)

Explanation: y components of the vector are given by:
RPY = 4 (4)2 = 64
RQY = 4 (5)2 = 100
Therefore, RP = 4i + 64j
RQ = 5i + 100j
Now, position difference from point P to point Q if given by
RPQ = RP – RQ = (4-5) i + (64 – 100) j = – i – 36j = – (i + 36j)

10. Two vectors are given by: Rp = 13i + 17j and RQ = 20i + 11j. Position difference from point P to point Q is given by ______
a) 6 ∠ 49.4°
b) 7 ∠ 40.6°
c) 9.23 ∠-40.6°
d) 9.23 ∠ 40.6°

Explanation: RPQ = RP – RQ = (13-20) i + (17-11) j = -7i + 6j
|RPQ| = $$\sqrt{(-7)^2+6^2}$$ = 9.23
Tanˉ($$\frac{6}{-7}$$) = -40.6°

11. The path of moving point is defined by the equation z=x2-y. Find the position difference from point P to point Q. If RPX = 1, RQX = 3, RPz= 2 and RQz= 4.
a) -2i – 6j – 2k
b) 2i + 6j + 2k
c) 4i + 4j + 6k
d) -6i + 2j – 2k

Explanation: y components of the vector are given by:
RPY = (1)2 – 2 = -1
RQY = (3)2 – 4 = 5
Therefore, RP = i – j + 2k
RQ = 3i + 5j + 4k
Now, position difference from point P to point Q if given by
RPQ = RP – RQ = (1-3) i + (-1-5) j + (2-4) = -2i – 6j – 2k

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