Vector Space: Overview, Questions, Preparation

Vectors are physical quantities that have magnitude as well as direction. These vectors’ magnitudes are known as scalars, which have no classified direction and can be summed up or removed just as dealt with by generic mathematical equations.

A space composed of vectors, collectively with the associative and commutative law of the addition of vectors and the associative and distributive process of multiplication of vectors by scalars, is called vector space.

The basic properties of the vector space are:

• The addition operation of a finite list of vectors v1 v2, . . , vk can be calculated in any order, then the solution of the addition process will be the same.
• If x + y = 0, then the value should be y = −x.
• The negative value of 0 is 0. This means that the value of −0 = 0.
• The negation or negative value of a vector’s negation’s negative value is the vector itself: −(−v) = v.
• If x + y = x, if and only if y = 0. Therefore, 0 is the only vector that behaves like 0.
• The product of any vector with zero times gives the zero vector. 0 x y = 0 for every vector in y.
• For every real number c, any scalar times the zero vector is the zero vector. c0 = 0
• If the value cx= 0, then either c = 0 or x = 0. The product of a scalar and a vector is equal to zero when either a scalar is 0, or a vector is 0.
• The scalar value −1 times a vector is the vector’s negation: (−1)x = −x. We define subtraction in terms of addition by defining x − y as an abbreviation for x + (−y).

This topic is taught in detail in 12th Standard Mathematics and constitutes on an average 14 marks in the board examination, often integrated with 3-D geometry. Apart from this, it finds massive application in higher Mathematics and Physics problems. All of the higher subjects are founded on the basic principles taught about vectors.

1. Classify the following measures as scalars and vectors.

(i) 10 kg                    (ii) 2 meters north-west                            (iii) 40° 
(iv) 40 watt               (v) 10–19 coulomb                                     (vi) 20 m/s2
Solution.
(i) Scalar               (ii) Vector            (iii) Vector
(iv) Scalar              (v) Vector           (vi) Vector

2. Classify the following as scalar and vector quantities.
(i) time period     (ii) distance     (iii) force      (iv) velocity      (v) work done
Solution.
(i) Vector    (ii) Scalar    (iii) Vector    (iv) Vector     (v) Scalar

3. Answer the following as true or false.
(i) a and –a are collinear.
(ii) Two collinear vectors are always equal in magnitude.
(iii) Two vectors having the same magnitude are collinear.
(iv) Two collinear vectors having the same magnitude are equal.
Solution
(i) True
(ii) False
(iii) False
(iv) False

Q: What is the definition of a vector?

Q: What is a unit vector?

Q: What is a zero vector?

Q: How do you differentiate scalars and vectors?

Q: How are vectors and vector space different?