Vectors and scalars, direction cosines, addition, components, section formula, and the scalar (dot) and vector (cross) products with their uses: angles, projections, perpendicularity and areas. Our own step-by-step solutions to Exercises 10.1 to 10.4 and the Miscellaneous Exercise, one page per exercise, set out for step marks.
Each exercise has its own page with every question solved step by step. Pick the one you are on.
What it tests. A scalar has magnitude only; a vector has magnitude and direction. Equal vectors: same magnitude and same direction (wherever they start). Collinear vectors lie along parallel lines, in the same or opposite direction. Coinitial vectors start at the same point.
What it tests. For \(\vec r = x\hat i + y\hat j + z\hat k\): \(|\vec r| = \sqrt{x^2 + y^2 + z^2}\), unit vector \(\hat r = \dfrac{\vec r}{|\vec r|}\), direction cosines \(\dfrac{x}{|\vec r|}, \dfrac{y}{|\vec r|}, \dfrac{z}{|\vec r|}\). \(\overrightarrow{PQ}\) = (position vector of Q) − (position vector of P). Section formula: \(\dfrac{m\vec b + n\vec a}{m + n}\) (internal), \(\dfrac{m\vec b - n\vec a}{m - n}\) (external).
What it tests. \(\vec a \cdot \vec b = |\vec a||\vec b|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3\). Perpendicular exactly when the dot product is 0. Projection of \(\vec a\) on \(\vec b\): \(\dfrac{\vec a \cdot \vec b}{|\vec b|}\). \(\vec a \cdot \vec a = |\vec a|^2\), so expand \((\vec a + \vec b) \cdot (\vec a + \vec b)\) like an algebraic square.
What it tests. \(\vec a \times \vec b = \begin{vmatrix}\hat i & \hat j & \hat k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3\end{vmatrix}\), perpendicular to both, with \(|\vec a \times \vec b| = |\vec a||\vec b|\sin\theta\). Parallel (collinear) vectors have zero cross product. Area of parallelogram \(= |\vec a \times \vec b|\); of triangle ABC \(= \tfrac12|\overrightarrow{AB} \times \overrightarrow{AC}|\).
What it tests. Mixed practice with components, unit vectors, section formula, dot and cross products; write every vector in \(\hat i, \hat j, \hat k\) form first.
Vector Algebra has 39 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.
Also useful: free MCQs and case studies for Vector Algebra · Class 12 formula sheet · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan
Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.