Vector Algebra
Class 12 · Mathematics
4 ideas in Class 12 Mathematics, explained in plain words, with games and quizzes to practise.
Vectors, Types and Scalars
Part of: Vectors and Their Arithmetic
Vectors carry magnitude plus direction, while scalars carry magnitude alone. Zero vectors lack direction, unit vectors measure one, and equal vectors match in size and heading anywhere. Direction cosines square-sum to one, and ratios scale into them.
Why it matters. Physics forces, velocities and fields all speak vector, and types gate every operation.
Formulas
- Direction cosines square-sum to one
- Magnitude is root of squared components
- Equal vectors match in size and heading
Key words
- Vector
- : A quantity with magnitude plus direction.
- Scalar
- : A quantity with magnitude alone.
- Zero vector
- : A vector with zero magnitude and no fixed direction.
- Unit vector
- : A vector with magnitude exactly one.
- Direction cosines
- : Cosines of angles with the three axes, squaring to one.
- Collinear vectors
- : Vectors parallel to one line, whatever their sizes.
Try it in two minutes
- Sort five measures into scalar and vector piles.
- Check two arrows for equal size and heading.
- Verify direction cosines square-sum to one.
Try at home. Name the vector type of a 5 km arrow pointing south.
Addition, Scalars and Sections
Part of: Vectors and Their Arithmetic
Vectors add nose-to-tail, matching components sum directly. Scalars stretch every component, and dividing by length builds unit vectors. Section formula weights endpoints by opposite ratios, with midpoints halving the sum.
Why it matters. Resultant forces, journeys and midpoints all compute this way.
Formulas
- Add matching components directly
- Unit vector is vector over its length
- Midpoint halves the position sum
Key words
- Resultant
- : The single sum of two or more vectors.
- Triangle law
- : Nose-to-tail addition giving the third side.
- Scalar multiple
- : Every component stretched by one number.
- Section formula
- : Ratio-weighted endpoints for division points.
Try it in two minutes
- Add two arrows nose-to-tail on paper.
- Stretch one arrow by 3 and observe.
- Halve a segment sum for its midpoint.
Try at home. Find the midpoint of (4, 6, 8) and (2, 0, 4).
Dot Product and Angles
Part of: Dot and Cross Products
The dot product multiplies matching parts and sums them, equalling lengths times cosine of the angle. Zero marks perpendicular pairs, the sign reads acute versus obtuse, and dividing by a length projects one vector onto another.
Why it matters. Work, angles and shadows in physics all compute through dots.
Formulas
- Dot is sum of matching-part products
- Zero dot marks perpendicular pairs
- Projection divides dot by length
Key words
- Dot product
- : The scalar sum of matching-part products.
- Perpendicular test
- : Zero dot product for nonzero vectors.
- Projection
- : The shadow length of one vector on another.
- Angle formula
- : Cosine as dot over length product.
Try it in two minutes
- Multiply matching parts of two arrows.
- Read zero as a right angle.
- Project by dividing with a length.
Try at home. Find the angle between (2, 0, 0) and (1, 1, 0).
Cross Product and Areas
Part of: Dot and Cross Products
The cross product stands perpendicular to both factors with length equal to the parallelogram they span. Parallel pairs cross to zero, swapping factors flips the sign, and halving gives triangle areas.
Why it matters. Torque, normals and areas in physics all compute through crosses.
Formulas
- Cross length is product times sine
- Parallel pairs cross to zero
- Triangle area halves the cross length
Key words
- Cross product
- : The perpendicular vector with parallelogram length.
- Right-hand rule
- : Curling fingers from first to second for direction.
- Parallelogram area
- : The length of the cross product.
- Anticommutativity
- : Swapping factors flips the cross sign.
Try it in two minutes
- Point fingers along both factors.
- Curl from first to second for direction.
- Weigh the spanned parallelogram.
Try at home. Compute (1, 0, 0) × (0, 1, 0) and name the result.
What your child will practise
- Addition, Scalars and Sections: Jungle Puzzle (Clue Puzzle)
- Addition, Scalars and Sections: Space Quick Fire (Quick Fire)
- Cross Product and Areas: Candy Puzzle (Clue Puzzle)
- Dot Product and Angles: Jungle Quest (Flip and Match)
- Vectors, Types and Scalars: Candy Quick Fire (Quick Fire)
- Quiz: Addition, Scalars and Sections (8 questions)
- Quiz: Cross Product and Areas (8 questions)
- Quiz: Dot Product and Angles (8 questions)
- Quiz: Vectors, Types and Scalars (8 questions)