Scalars and Vectors

• Very Important Multiple Choice Questions (MCQs) with Answers of the Chapter No. 2: Scalars and Vectors in Physics Class 11 Sindh Book Boards.
• Definition of Scalars with Examples and Properties
• Direct Preference, Arithmetical Operations, Representation, Equivalence.
• Definition of Vectors with Examples and Properties
• Direct Preference, Arithmetical Operations, Representation, Equivalence.
• Definition of Resolution of vectors
• Definition of Rectangular components (i) The Horizontal component on X-Component (ii) The Vertical component on Y- Component.
• Definition of unit vector with Notation and Mathematically
• Definition of position vector with Notation
• Definition of Free Vector
• Definition of Null Vector
• Three different ways. (i) By head - to - tail rule, or graphical method, or triangular law of vector addition. (ii) Analytical method. (iii) Addition of vectors by rectangular components method.
• Describe a graphical method (Head to Tail rule)
• Properties of vector addition: (i) community law of vector addition (ii) associative law of vector addition (iii) Analytical Method of Vector Addition (iv)  Rectangular Components Method
• Definition of a scalar product with examples and representation and explanations.
• characteristic of dot product
• Definition of a Vector product with examples and representation and explanations.
• Definition of torque with formula
• definition for Force with Formula
• Direction of vector product
• Characteristics of vector product: (i) commutative law (ii) distributive law
•  The following forces act on a particle P: F1 = 2i + 3j – 5k, F2 = -5i + j + 3k, F3 = i – 2j + 4k, F4 = 4i – 3j – 2k, measured in newtons Find (a) the resultant of the force (b) the magnitude of the resultant force. 
•  If A = 3i – j – 4k, B = -2i + 4j – 3k and C = i + 2j – k, find (a) 2A – B + 3C, (b) │A + B + C│, (c) │3A – 2B + 4C│, (d) a unit vector parallel to 3A – 2B + 4C
• Two tugboats are towing a ship. Each exerts a force of 6000N, and the angle between the two ropes is 60o . Calculate the resultant force on the ship.
• The position vectors of points P and Q are given by r1 = 2i +3j- k, r2 = 4i – 3j + 2k. Determine 𝑷 𝑸 in terms of rectangular unit vector i, j and k and find its magnitude.
• Prove that the vectors A = 3i + j – 2k, B = -i + 3j + 4k and C = 4i – 2j – 6k can form the sides of a triangle. Find the length of the medians of the triangle
• Find the rectangular components of a vector A, 15 unit long when it form an angle with respect to +ve x-axis of (i) 50o , (ii) 130o (iii) 230o , (iv) 310o .
• Two vectors 10 cm and 8 cm long form an angle of (a) 60o (b) 90o and (c) 120o . Find the magnitude of difference and the angle with respect to the larger vector.
• The angle between the vector A and B is 60o . Given that │A│=│B│ = 1, calculate (a) │B - A│; (b) │B + A│
• A car weighing 10,000 N on a hill which makes an angle of 20° with the horizontal. Find the components of car’s weight parallel and perpendicular to the road.
• Find the angle between A = 2i + 2j – k and B = 6i – 3j + 2k.
• Find the projection of the vector A = i – 2j + k onto the direction of vector B = 4i – 4j + 7k.
• Find the angles α, β, γ which the vector A = 3i – 6j + 2k makes with the positive x, y, z axis respectively
• Find the work done in moving an object along a vector r = 3i + 2j – 5k if the applied force is F = 2i – j – k.
• Find the work done by a force of 30,000 N in moving an o bject through a distance of 45 m when: (a) the force is in the direction of motion: and (b) the force makes an angle of 40o to the direction of motion. Find the rate at which the force is working at a time when the velocity is 2m/s.
• Two vectors A and B are such that │A│= 3, │B│= 4, and A.B = -5, find (a) the angle between A and B (b) the length │A + B│ and │A - B│ (c) the angle between (A + B) and (A – B)
•  If A = 2i – 3j – k, B = i + 4j – 2k. Find (a) A x B (b) B x A (c) (A + B) x (A – B)
• Determine the unit vector perpendicular to the plane of A = 2i – 6j – 3k and B = 4i + 3j – k.
• Using the definition of vector product, prove the law of sines for plane triangles of sides a, b and c.
• If r1 and r2 are the position vectors (both lie in xy plane) making angle θ1 and θ2 with the positive xaxis measured counter clockwise, find their vector product when (i) │r1│= 4 cm θ1 = 30o │r2│= 3 cm θ2 = 90o (ii) │r1│= 6 cm θ1 = 220o │r2│= 3 cm θ2 = 40o (iii) │r1│= 10 cm θ1 = 20o │r2│= 9 cm θ2 = 110o.
• Determine the unit vector perpendicular in the plane of A= 2i-6j-3k and B =4i+3j-k
• Two vectors A and B are such that |A|=4, |B|=6 and A.B=13.5.Find the magnitude of difference of vectors and angle between A and B.
• If the vector 𝑨 = 𝒂𝒊 + 𝒋 − 𝟐𝒌 and 𝑩 = 𝒊 + 𝒂𝒋 + 𝒌 are perpendicular to each other then find the value ‘a’.
• Determine the unit vector perpendicular to the plane of A = 3i + 4j -k and B =4i+3j-2k vectors.
• Two sides of a triangle are formed by vectors A =3i + 6j -2k and B = 4i - j + 3k . Determine the area of the triangle.
• Determine the unit vector perpendicular to the plane containing A and B. 𝑨 = 𝟐𝒊 − 𝟑𝒋 − 𝒌 and 𝑩 = 𝒊 + 𝟒𝒋 − 𝟐 𝒌
• If 𝑨 = 𝟑𝒊 + 𝒋 − 𝟐𝒌 and 𝑩 = −𝒊 + 𝟑𝒋 + 𝟒𝒌 . Find the projection of A on to B.
• Two vectors A and B are such that A=4 , B=6 and |A-B|=5. Find |A+B|
• If one of the rectangular components of force 50 N is 25N; find the value of the other.