Chapter 4: Vector AlgebraClass 12 Mathematics — summary, notes, extra questions & MCQ quiz

Summary

A vector has both magnitude and direction, in contrast to a scalar. This chapter introduces position vectors, direction cosines and direction ratios, and the representation \(\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\). Vectors are added by the triangle and parallelogram laws and scaled by real numbers. Two products are central: the scalar (dot) product \(\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta\), which measures projection and tests perpendicularity, and the vector (cross) product \(\vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\sin\theta\,\hat{n}\), whose magnitude gives the area of the parallelogram spanned by the vectors and whose direction is perpendicular to both. The unit vector \(\hat{a}=\dfrac{\vec{a}}{|\vec{a}|}\) and the section formula are also covered. These tools underpin three-dimensional geometry and physics.

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