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Vector Algebra: CBSE Class 12 Maths knowledge organiser

Everything to know about vector algebra on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Vector
A quantity with magnitude and direction.
Scalar product
a · b = |a||b|cos θ, a number; zero for perpendicular vectors.
Vector product
a × b, a vector perpendicular to both, of magnitude |a||b|sin θ.

Key formulas

Magnitude; unit vector\(|\vec a|=\sqrt{x^2+y^2+z^2},\) \(\hat a=\frac{\vec a}{|\vec a|}\)
Direction cosines\(l=\tfrac x{|\vec r|},\ m=\tfrac y{|\vec r|},\ n=\tfrac z{|\vec r|},\) \(l^2+m^2+n^2=1\)
\(\overrightarrow{PQ}\)\((x_2-x_1)\hat i+(y_2-y_1)\hat j+(z_2-z_1)\hat k\)
Section, \(m:n\) internal; external\(\frac{m\vec b+n\vec a}{m+n};\) \(\frac{m\vec b-n\vec a}{m-n}\)
Dot product\(\vec a\cdot\vec b=|\vec a||\vec b|\cos\theta=a_1b_1+a_2b_2+a_3b_3\)
Perpendicular; projection of \(\vec a\) on \(\vec b\)\(\vec a\cdot\vec b=0;\) \(\frac{\vec a\cdot\vec b}{|\vec b|}\)
Cross product\(\vec a\times\vec b=|\vec a||\vec b|\sin\theta\,\hat n=\begin{vmatrix}\hat i&\hat j&\hat k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}\)
Area: parallelogram; triangle\(|\vec a\times\vec b|;\) \(\tfrac12|\vec a\times\vec b|\)
Parallelogram from its diagonals\(\tfrac12|\vec d_1\times\vec d_2|\)
Parallel \(\vec a\times\vec b=\vec 0\); \(\vec a\times\vec b=-\vec b\times\vec a\); \(\hat i\times\hat j=\hat k\), \(\hat j\times\hat k=\hat i\), \(\hat k\times\hat i=\hat j\)

Worked example

\(|\vec{a}| = 2\), \(|\vec{b}| = 3\), \(\vec{a}\cdot\vec{b} = 3\): \(|\vec{a} - \vec{b}|^2 = 4 + 9 - 6 = 7\), so \(|\vec{a} - \vec{b}| = \sqrt7\); \(\cos\theta = \tfrac12\), \(\theta = \tfrac\pi3\).

Common mistakes

  • Middle term of the cross-product determinant: it is \(-\hat{j}(a_1b_3 - a_3b_1)\).
  • Using \(\tfrac{m\vec{a} + n\vec{b}}{m + n}\) (weights swapped) in the section formula.
  • Giving only one sign for a unit perpendicular vector, or the wrong one when a condition (acute/obtuse with an axis) is given.
  • Confusing the scalar projection (a number) with the projection vector.

You should be able to…

  • Find magnitudes, unit vectors, direction cosines, section-formula points and simple dot/cross products.
  • Use dot and cross products for angles, projections, perpendicularity, collinearity and areas at 2- and 3-mark level.
  • Write complete multi-step vector solutions (triangle and parallelogram problems, case studies with forces or displacements).
  • Work with vectors given only by magnitudes and angles, minimisation and perpendicular-component arguments.

The printable sheet

Vector Algebra knowledge organiser for CBSE Class 12 Maths: one A4 page of key definitions, formulas, a worked example and common mistakes
Vector Algebra knowledge organiser (CBSE Class 12 Maths), A4. Download the PDF.

Revise it next

Other CBSE Class 12 Maths topics: Relations and Functions · Inverse Trigonometric Functions · Matrices · Determinants · Continuity and Differentiability · Application of Derivatives · Integrals · Application of Integrals · Differential Equations · Three Dimensional Geometry · Linear Programming · Probability · All CBSE Class 12 Maths organisers