Vector Calculus: Overview, Questions, Preparation

A quantity that has magnitude, as well as direction, is called a vector. Point A from where the vector starts is called its initial point, and the point B where it ends is called its terminal point. The distance between the initial and terminal points of a vector is called the magnitude (or length) of the vector. The arrow indicates the direction of the vector. 

Addition of Vectors

Let a and b be any two vectors. From the terminal point of a vector, b is drawn. The vector from the initial point O of A to a terminal point B of b is called the sum of vectors a and b. This sum is denoted by a + b. This is called the triangle law of the addition of vectors.

Parallelogram Law

Let a and b be any two vectors. From the initial point of a vector b is drawn and a parallelogram
OACB is completed with OA and OB as adjacent sides. The vector OC is defined as the sum of
a and b. This is called the parallelogram law of the addition of vectors. The sum of two vectors is also called their resultant and the process of addiction as composition.

Multiplication of a Vector by a Scalar

Let a be a given vector and λ be a scalar. Then, the product of the vector a by the scalar λ is λ.a. This is called the multiplication of vectors by the scalar.

The Position Vector of a Point 

The position vector of a point P concerning a fixed point, say O, is the vector OP. The fixed point is called the origin. Let PQ be any vector, then,

PQ = PO + OQ = — OP + OQ = OQ — OP = Position vector of Q — Position vector of P.

Vector or Cross Product of two Vectors

The vector product of the vectors a and b is denoted by a * b and it is defined as a * b = (|a| |b| sin θ) n = ab sin θ n …..(i). Here, a = |a|, b= |b|, θ is the angle between the vectors a and b. Also, n is a unit vector perpendicular to both a and b, such that a, b and n form a right-handed triad of vectors.

This section has the highest weightage in Class 12th Mathematics. The Calculus unit carries 35 marks, and the Vectors and 3-D Geometry carries 14 marks. 

1. Represent graphically a displacement of 40 km, 30° west of south.

Solution.

The vector OP is the required placement. 

2. Classify the following measures as scalars and vectors:
(i) 5 seconds (ii) 1000 cm³ (iii) 10 Newton (iv) 30 km/hr (v) 10 g/cm³ (vi) 20 m/s towards north

Solution. (i) Time-scalar (ii) Volume-scalar (iii) Force-vector (iv) Speed-scalar (v) Density-scalar (vi) Velocity-vector

3 Find the vector joining the points P(2, 3, 0) and Q(– 1, – 2, – 4) directed from P to Q.

Solution. Since the vector is to be directed from P to Q, clearly P is the initial point and Q is the terminal point. So, the required vector joining P and Q is the vector PQ, given by

Image source: NCERT

Q: What is the distributive law for three vectors?

Q: What is a scalar triple product?

Q: What is the reciprocal system of vectors?

Q: Define a linear combination of vectors. 

Q: What is the resolution of the components of a vector in a plane?