Kidz Olympiad馃帀

Mathematics Class 12 - Vector Algebra Notes

Comprehensive study notes for Class 12 - Vector Algebra olympiad preparation

Vector Algebra

Welcome to the chapter on Vector Algebra for Class 12. In this chapter, you will learn about vectors, their properties, operations on vectors, and their applications in geometry and physics. By the end of this chapter, you will be able to solve problems involving vectors and understand their importance in mathematics.

Key Concepts

  • Vector: A quantity that has both magnitude and direction.
  • Scalar: A quantity that has only magnitude.
  • Position Vector: A vector that represents the position of a point with respect to the origin.
  • Zero Vector: A vector with zero magnitude and no direction.
  • Unit Vector: A vector with magnitude 1, showing direction only.

Types of Vectors

  • Collinear Vectors: Vectors that are parallel to the same line.
  • Equal Vectors: Vectors with the same magnitude and direction.
  • Negative of a Vector: A vector with the same magnitude but opposite direction.

Algebra of Vectors

  • Addition: The sum of two vectors is found by joining them head to tail.
  • Subtraction: Subtracting a vector is the same as adding its negative.
  • Multiplication by a Scalar: Changes the magnitude of the vector but not its direction (unless the scalar is negative).

Section Formula

If a point divides the line segment joining two points A and B in the ratio m:n, then the position vector of the point is:

P = (mB + nA) / (m + n)

Scalar (Dot) Product

The scalar product of two vectors a and b is given by:

a 路 b = |a||b|cos胃

  • It is a scalar quantity.
  • If vectors are perpendicular, their dot product is zero.

Vector (Cross) Product

The vector product of two vectors a and b is given by:

a 脳 b = |a||b|sin胃 n虃

  • It is a vector quantity.
  • Direction is given by the right-hand rule.
  • If vectors are parallel, their cross product is zero.

Applications of Vectors

  • Finding distance and displacement.
  • Solving geometry problems (like area of triangle, parallelogram).
  • Physics problems involving force, velocity, and acceleration.

Practice Questions

  1. Find the unit vector in the direction of a = 3i + 4j.
  2. If a = 2i + 3j + k and b = i - 2j + 2k, find a + b.
  3. Calculate the dot product of a = i + 2j and b = 2i + j.
  4. If two vectors are perpendicular, what is their dot product?
  5. Find the area of a triangle with vertices at A(1,2,3), B(4,5,6), and C(7,8,9) using vectors.

Challenge Yourself

  • Prove that the diagonals of a parallelogram bisect each other using vectors.
  • If a and b are unit vectors and the angle between them is 60掳, find a 路 b and |a 脳 b|.

Did You Know?

  • Vectors are used in computer graphics to create animations and games!
  • Engineers use vectors to design bridges and buildings.

Glossary

  • Magnitude: The length or size of a vector.
  • Direction: The way in which a vector points.
  • Unit Vector: A vector with magnitude 1.
  • Dot Product: The scalar product of two vectors.
  • Cross Product: The vector product of two vectors.

Answers to Practice Questions

  1. Unit vector = (3i + 4j)/5 = 0.6i + 0.8j
  2. a + b = (2+1)i + (3-2)j + (1+2)k = 3i + 1j + 3k
  3. a 路 b = (1脳2) + (2脳1) = 2 + 2 = 4
  4. Zero
  5. Area = (1/2)|AB 脳 AC|, where AB = (3,3,3), AC = (6,6,6), so AB 脳 AC = 0, Area = 0 (points are collinear)

Practice vector algebra to solve geometry and physics problems with confidence!

Quick Navigation