10. Vector Algebra

10 topics4 h

Chapter 10 of NCERT Mathematics for Class 12: Vector Algebra. Three sections — Quantities that carry a direction, Adding and scaling and Two ways to multiply two vectors. Ten videos, 4 h in all.

Worked answers to the 73 exercise questions in this chapter: Exercise 10.1 · Exercise 10.2 · Exercise 10.3 · Exercise 10.4 · Miscellaneous Exercise

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Part 1

Quantities that carry a direction

19 minWhy some measurements need a direction attached and others do not19 min · 17 parts16 minMagnitude, the three angles with the axes, and the cosines and ratios they give16 min · 17 parts21 minThe named kinds — zero, unit, coinitial, collinear, equal, negative and free21 min · 19 parts

Part 2

Adding and scaling

19 minTwo laws for adding, why they agree, and what addition obeys19 min · 19 parts23 minStretching by a scalar, and dividing a vector by its own length23 min · 18 parts22 minSplitting a vector along the axes so the algebra becomes coordinate arithmetic22 min · 19 parts21 minPosition vectors, the vector joining two points, and dividing a segment in a ratio21 min · 19 parts

Part 3

Two ways to multiply two vectors

27 minA product that returns a number, and the angle you can extract from it27 min · 20 parts17 minProjection: how much of one vector points along another17 min · 14 parts26 minA product that returns a vector, where it points, and the area it measures26 min · 17 parts

What you will be able to do

  • Separate a measurement that is complete once a number and a unit are given from one that is not complete until a direction is added
  • Say what makes a vector a position vector, and what it is measured against
  • Recite the chapter's seven named kinds and say which page defines each
  • State the triangle law as an equation between three directed segments and read it off a drawing
  • State what the chapter asserts about the product of a vector and a scalar, and distinguish the assertions from the one thing it derives
  • Name the three unit vectors along the axes and say which three points define them
  • Derive the vector joining two points from their two position vectors using the triangle law
  • Explain the analogy the chapter uses to justify defining two products
  • Describe the projection vector of a vector on a directed line, in magnitude and in direction
  • Explain the right-handed convention using either of the chapter's two pictures

What this chapter assumes you already know

  • Distance, speed, area, volume and mass as measured quantities, from Class XI
  • Compass bearings, and reading an angle measured from a named direction
  • A line, a ray and a line segment, and the difference between them
  • Absolute value of a real number, and the fact that a length cannot be negative
  • A right handed system of rectangular coordinates in space, from Class XI, at the level of "a point carries three coordinates"
  • The distance formula in three dimensions, from Class XI
  • A right handed system of rectangular coordinates in space, from Class XI
  • The distance of a point from the origin in three dimensions, from Class XI

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "A vector is a quantity that has a direction, so anything with a direction is a vector."
  • "Distance and displacement are the same thing measured differently."
  • "Speed is scalar and velocity is vector, so the units must differ."
  • "If no direction is written, it must be a scalar."
  • "A directed line is just a line with an arrow drawn anywhere on it."
  • "The magnitude comes from the arrow."
  • "The modulus bars around a vector are just decoration, so I can compare them however I like."
  • "A vector belongs at the place it is drawn."