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CBSE Class 12 Maths

Vector Algebra

Each concept is a short lesson, then questions until you get three right.

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8 concepts, a short lesson each

  • Vectors0/4
  • Dot product0/2
  • Cross product0/2

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Vectors in space and their magnitude

a⃗=xi^+yj^+zk^\vec a = x\hat i + y\hat j + z\hat k has ∣a⃗∣=x2+y2+z2|\vec a| = \sqrt{x^2 + y^2 + z^2}; AB→\overrightarrow{AB} = position vector of B minus that of A; a unit vector is a⃗/∣a⃗∣\vec a/|\vec a|

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Vectors0 of 4

  1. Vectors in space and their magnitudea⃗=xi^+yj^+zk^\vec a = x\hat i + y\hat j + z\hat k has ∣a⃗∣=x2+y2+z2|\vec a| = \sqrt{x^2 + y^2 + z^2}; AB→\overrightarrow{AB} = position vector of B minus that of A; a unit vector is a⃗/∣a⃗∣\vec a/|\vec a|Start
  2. Adding vectors and the triangle lawAdd and subtract component by component; around a closed figure the vectors add to 0⃗\vec 0; scalar multiples are parallelStart
  3. Collinear and coplanar vectorsThree points are collinear when AB→=k AC→\overrightarrow{AB} = k\,\overrightarrow{AC}; a vector is a linear combination xa⃗+yb⃗+zc⃗x\vec a + y\vec b + z\vec c of three non-coplanar onesStart
  4. Section formulaInternal division in m:nm : n: mb⃗+na⃗m+n\dfrac{m\vec b + n\vec a}{m + n}; external: mb⃗−na⃗m−n\dfrac{m\vec b - n\vec a}{m - n}; midpoint and centroid as special casesStart

Dot product0 of 2

  1. Dot product and the angle between vectorsa⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ=a1b1+a2b2+a3b3\vec a\cdot\vec b = |\vec a||\vec b|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3; zero means perpendicularStart
  2. Projection of a vectorThe scalar projection of a⃗\vec a on b⃗\vec b is a⃗⋅b⃗∣b⃗∣\dfrac{\vec a\cdot\vec b}{|\vec b|}; the vector projection multiplies it by b^\hat bStart

Cross product0 of 2

  1. Cross producta⃗×b⃗\vec a\times\vec b is the determinant with rows i^ j^ k^\hat i\ \hat j\ \hat k, a⃗\vec a, b⃗\vec b; it is perpendicular to both, and b⃗×a⃗=−a⃗×b⃗\vec b\times\vec a = -\vec a\times\vec bStart
  2. Areas and unit normals from the cross product∣a⃗×b⃗∣=∣a⃗∣∣b⃗∣sin⁡θ|\vec a\times\vec b| = |\vec a||\vec b|\sin\theta is the parallelogram's area; half of it for a triangle; ±a⃗×b⃗∣a⃗×b⃗∣\pm\dfrac{\vec a\times\vec b}{|\vec a\times\vec b|} are the unit normalsStart

Practise Vector Algebra

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answersPast board questionsAsked from this chapter, with solutions