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Cheat sheet · Class 12 · CBSE · Commerce Mathematics

Vector Algebra cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

CBSE Class 12 · Mathematics

Cheat sheet

Chapter 10Vector Algebra

Dot product → a number (cos); cross product → a vector (sin).

1. Basics

  • a⃗=a1i^+a2j^+a3k^\vec a = a_1\hat i + a_2\hat j + a_3\hat k, ∣a⃗∣=a12+a22+a32|\vec a| = \sqrt{a_1^2 + a_2^2 + a_3^2}.
  • Direction cosines l, m, n: l² + m² + n² = 1.
  • Section formula: mb⃗+na⃗m+n\dfrac{m\vec b + n\vec a}{m + n}.

2. Products

a⃗⋅b⃗=∣a∣∣b∣cos⁡θ=a1b1+a2b2+a3b3\vec a\cdot\vec b = |a||b|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3
∣a⃗×b⃗∣=∣a∣∣b∣sin⁡θ|\vec a\times\vec b| = |a||b|\sin\theta
a⃗×b⃗=∣i^j^k^a1a2a3b1b2b3∣\vec a\times\vec b = \begin{vmatrix}\hat i & \hat j & \hat k\\ a_1 & a_2 & a_3\\ b_1 & b_2 & b_3\end{vmatrix}
  • Projection of a⃗\vec a on b⃗\vec b: a⃗⋅b⃗∣b⃗∣\dfrac{\vec a\cdot\vec b}{|\vec b|}.
  • Area of parallelogram =∣a⃗×b⃗∣= |\vec a\times\vec b|; triangle =12∣a⃗×b⃗∣= \tfrac12|\vec a\times\vec b|.

3. Key Points

  • Perpendicular: a · b = 0.
  • Parallel: a × b = 0.
  • a × b = −(b × a).
  • î · î = 1, î × ĵ = k̂.

Remember: Magnitude | dot vs cross | projection | areas

Vector Algebra: questions and notes

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