Vector Algebra
Class 12 Mathematics · Mathematics Part II · Chapter 4
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Mathematics
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Physics, Chemistry, Maths + English
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Physics, Chemistry, Biology + English
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Accountancy, Business Studies, Economics + English
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Mathematics
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Mathematics
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Physics
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History
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Hindi Elective
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Sociology
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Economics
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Urdu Elective
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Geography
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Accountancy
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English Core
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Chemistry
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Business Studies
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Hindi Core
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हिंदी (आधार)
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Accountancy
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اردو (اختیاری)
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हिंदी (आधार)
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संस्कृतम् (केन्द्रिकम्)
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Physics
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हिंदी (ऐच्छिक)
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संस्कृत (ऐच्छिक)
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Physics
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हिंदी (ऐच्छिक)
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اردو (کور)
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Chemistry
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Urdu Elective
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Computer Science
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Chemistry
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Fine Art · Painting
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हिंदुस्तानी संगीत (गायन / वादन)
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Biology
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Informatics Practices
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Accountancy
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Biotechnology
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Accountancy
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सृजनात्मक लेखन और अनुवाद
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Business Studies
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हिंदुस्तानी संगीत (तबला / पखावज)
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Business Studies
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Economics
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Economics
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History
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History
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History
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Geography
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Geography
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Geography
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English Elective · Kaleidoscope
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Political Science
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Political Science
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Sociology
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Sociology
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Psychology
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English Core · Flamingo
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English Core · Vistas
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Home Science
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Home Science
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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.
Vectors and scalarsPosition vectors and direction cosinesAddition and scalar multiplicationScalar (dot) productVector (cross) product
Key terms
Vector
A quantity with both magnitude and direction.
Position vector
The vector from the origin to a point.
Unit vector
\(\hat{a}=\dfrac{\vec{a}}{|\vec{a}|}\), a vector of magnitude one.
Dot product
\(\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta\), a scalar.
Cross product
\(\vec{a}\times\vec{b}=|\vec{a}||\vec{b}|\sin\theta\,\hat{n}\), a vector.
Direction cosines
The cosines of the angles a vector makes with the axes.
Extra questions & answers
Try one from this chapter
A vector is a quantity having: