Ans. Unit vector: “A unit vector in a given direction is a vector with magnitude one in that direction”.
A=A/A
Position vector: “The position vector r is a vector that describes the location of a particle with respect to the origin”.
Components of a vector: “A component of a vector is its effective value in a given direction”.
or
Vectors into which the given vector is resolved are called components of the given vector.
Q.2 The vector sum of three vectors gives a zero resultant. What can be the orientation of the vectors?
Ans. If three vectors are drawn, to make a closed triangle, then their vector sum will be zero. As shown in the figure.
A+B+C=0
Q.3 Vector A lies in the xy plane. For what orientation will both of its rectangular
components be negative? For what orientation will its components have opposite
signs?
Ans.
- Vector A lies in 3rd quadrant, its rectangular components will be negative.
- When the vector will lie in 2nd or 4th quadrant, its components will have opposite signs.
Ans. No. Its magnitude cannot be zero, when one of the component of the vector is not zero. i.e. if
Ax ≠0 & Ay = 0
then
A = √ Ax2 + (0)2 = √ Ax2 = Ax ≠ 0
Q.5 Can a vector have a component greater than the vector’s magnitude?
Ans. A vector may have a component greater than the magnitude of the vector. But the rectangular components can never be greater than the vector's magnitude. It may be equal to any of its rectangular component but may not be greater.
Q.6 Can the magnitude of a vector have a negative value?
Ans. No. The magnitude of a vector has always positive values; A = √ Ax2 + Ay2, because the square of the values of Ax and Ay always be positive. so magnitude of A always be positive.
Q.7 If A + B = 0. What can you say about the components of the two vectors?
Ans.
A + B = 0
This represents that,
- Ax + Bx=0
- Ay + By=0
Q.8 Under what circumstances would a vector have components that are
equal in magnitude?
Ans. In case of rectangular components, the magnitude of the two components will be equal if the vector makes an angle of 45° with +x-axis.
Ans. In case of rectangular components, the magnitude of the two components will be equal if the vector makes an angle of 45° with +x-axis.
Q.9 Is it possible to add a vector quantity to a scalar quantity? Explain.
Ans. No. It is not possible to add a vector quantity to a scalar quantity, different physical quantities cannot be added.
Q.10 Can you add zero to a null vector?
Ans. No. We cannot add zero to a null vector. Because zero, a scalar quantity cannot be added with a vector quantity—the null vector, because both quantities are of different nature.
Q.11 Two vectors have unequal magnitudes. Can their sum be zero? Explain.
Ans. No,their sum can not be zero. The sum of two unequal vectors cannot be zero. For the sum of vectors to be zero, the vectors must have equal magnitude with opposite directions.
Q.12 Show that the sum and difference of two perpendicular vectors of equal lengths are also perpendicular and of the same length.
Ans. Let us suppose there are two vectors A and B which are equal in magnitude and perpendicular to each other.
According to head to tail rule,
Hence we can say that R is perpendicular to Ro. Therefore, both sum and difference are perpendicular to one another. Now by using the formula for magnitude of rectangular components, we have
Ans. If the angle between two vectors of same magnitude is 120º, the magnitude of their resultant vector is also the same.
This can be proved as follows;
Q.14 The two vectors to be combined have magnitudes 60 N and 35 N. Pick the correct
answer from those given and tell why is it the only one of the three that is correct.
i) 100 N ii) 70 N iii) 20 N
Ans. A1 = 60N and A2 = 35N Answer (ii) 70N is correct. For maximum value, both vectors in same direction,
A1 + A2 = 60 + 35 = 95 → cannot be (i) 100N, For minimum value, both vectors having opposite direction,
A1 - A2 = 60 + 35 = 25 → cannot be (iii) 20N
Q.15 Suppose the sides of a closed polygon represent vector arranged head to tail. What is the sum of these vectors?
Ans. The sides of a closed polygon represent vector arranged head to tail then their sum will be null vector.
A+B+C+D+E=0
Q.16 Identify the correct answer;
i) Two ships X and Y are travelling in different directions at equal speeds. The
actual direction of motion of X is due north but to an observer on Y, the apparent
direction of motion of X is north-east. The actual direction of motion of Y as
observed from the shore will be
(A) East (B) West (C) South-East (D) South-West
ii) A horizontal force F is applied to a small object P of mass M at rest on a smooth
plane inclined at an angle θ to the horizontal as shown in the figure. The
magnitude of the resultant force acting up and along the surface of the plane,
on the object is
F cos θ - mg sin θ
F sin θ - mg cos θ
F cos θ + mg cos θ
F sin θ + mg sin θ
mg tan θ
Ans. (i) The correct answer is (B) West.
Apparent direction of motion
Actual direction of X
Actual direction of Y (ii) The correct answer is (a) F cos θ - mg sin θ the resultant force acting Up and along the surface of the plane is = F cos θ - mg sin θ
Q.17 If all the components of the vectors, A1 and A2 were reversed, how would this alter
A1 x A2 ?
Ans. For A1 = - A1 & A2 = - A2 -A1 x -A2 = A1 x A2 There will be no effect, if all the components of the vectors A1 & A2 are reversed.
Q.18 Name the three different conditions that could make A1 x A2 = 0.
Ans. A1 x A2 could be zero,
if i) A1 is null vector ; 0 x A2 = 0
ii) A2 is null vector ; A1 x 0 = 0
iii) A1 x A2 are parallel or anti-parallel, i.e.
A1 x A2 = A1 A2 sin θ = A1 A2 sin 180o = 0
Q.19 Identify true or false statements and explain the reason.
a) A body in equilibrium implies that it is not moving nor rotating.
b) If coplanar forces acting on a body form a closed polygon, then the body is said to be in equilibrium.
Ans. a) It is false. A body moving with constant velocity can also be in equilibrium. b) It is true. The vector sum will be zero, for the coplanar forces forming a closed polygon, fulfils the 1st condition of equilibrium.
Q.20 A picture is suspended from a wall by two strings. Show by diagram the
configuration of the strings for which the tension in the strings will be minimum.
Ans. For T minimum, θ = 90o 2T=W
T=W/2 Tension will be the half of the weight.
T
Q.21 Can a body rotate about its centre of gravity under the action of its weight?
Ans. No. A body cannot rotate about its center of gravity under the action of its weight. Because moment arm will be zero, so torque or turning effect will be zero.
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