NCERT Class 12-Mathematics: Chapter – 10 Vector Algebra Part 1 (For CBSE, ICSE, IAS, NET, NRA 2022)
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10.1 Overview
10.1. 1 A quantity that has magnitude as well as direction is called a vector.
10.1. 2 The unit vector in the direction of is given by and is represented .
10.1. 3 Position vector of a point is given as and its magnitude as , where is the origin.
10.1. 4 The scalar components of a vector are its direction ratios, and represent its projections along the respective axes.
10.1. 5 The magnitude r, direction ratios and direction cosines of any vector are related as:
10.1. 6 The sum of the vectors representing the three sides of a triangle taken in order is
10.1. 7 The triangle law of vector addition states that “If two vectors are represented by two sides of a triangle taken in order, then their sum or resultant is given by the third side taken in opposite order” .
10.1. 8 Scalar multiplication
If is a given vector and a scalar, then is a vector whose magnitude is . The direction of is same as that of if is positive and, opposite to that of if is negative.
10.1. 9 Vector joining two points
If and are any two points, then
10.1. 10 Section formula
The position vector of a point R dividing the line segment joining the points P and Q whose position vectors are and
(i) In the ratio internally, is given by
(ii) in the ratio externally, is given by
10.1. 11 Projection of along is and the Projection vector of along is
10.1. 12 Scalar or dot product
The scalar or dot product of two given vectors and having an angle θ between them is defined as
10.1. 13 Vector or cross product
The cross product of two vectors and having angle θ between them is given as
where is a unit vector perpendicular to the plane containing and and , form a right-handed system?
10.1. 14 and are two vectors and is any scalar, then
Angle between two vectors and is given by
10.2 Solved Examples
Short Answer (S. A)
Question 1:
Find the unit vector in the direction of the sum of the vectors
Answer:
Let denote the sum of and We have
Now .
Thus, the required unit vector is
Question 2:
Find a vector of magnitude 11 in the direction opposite to that of where P and Q are the points and , respectively.
Thus
Therefore, unit vector in the direction of is given by
Hence, the required vector of magnitude 11 in direction of is