NDA Vector Algebra: Dot Product, Cross Product and Geometry

Owned by UPSC NDA Mathematics syllabus. Exact NDA-relevant scope: Vectors in two and three dimensions; magnitude and direction; unit and null vectors; vector addition; scalar multiplication; dot product; cross product; applications to work done by a force, moment of a force and geometrical problems.

Vectors in two and three dimensions; magnitude and direction; unit and null vectors; vector addition; scalar multiplication; dot product; cross product; applications to work done by a force, moment of a force and geometrical problems.

Coordinate geometry, Pythagoras, basic trigonometry and algebra.

• Scalar vs vector
• Magnitude
• Unit vector
• Dot product
• Cross product

• |a|=sqrt(a1^2+a2^2+a3^2)
• Unit vector along a: a/|a| for a!=0
• a.b=|a||b|cos theta
• |a x b|=|a||b|sin theta
• a.b=0 for perpendicular non-zero vectors
• a x b=0 for parallel non-zero vectors
• Work W=F.s
• Moment about origin can be represented by r x F

• Unit vector formula excludes the zero vector.
• Dot-product angle formula requires non-zero vectors.
• Cross-product direction follows right-hand convention.
• Parallel / perpendicular conclusions require attention to zero vectors.

1. For a=(3,4), |a|=5 and unit vector is (3/5,4/5).
2. For a=(1,0,0), b=(0,2,0), a.b=0 so they are perpendicular; a x b=(0,0,2).
3. A constant force (2,3) moving an object through displacement (4,1) does work 2x4+3x1=11 in consistent units.

• Using dot product where cross product is required.
• Forgetting cross-product order changes sign.
• Normalising the zero vector.
• Dropping square roots incorrectly in magnitude.
• Confusing scalar product result with a vector.

• Knowledge Gap: The concept, definition or theorem needed to solve the item is not understood.
• Recall Gap: The concept was learned but a formula, condition or relation cannot be retrieved accurately.
• Execution Error: The correct method is known but algebra, arithmetic, sign, substitution, diagram or step execution fails.
• Decision / Selection Error: The candidate chooses an inefficient method, continues a time-heavy path, or attempts a low-confidence item without adequate basis.
• Needs Review: Evidence is insufficient or contradictory; do not mislabel the error until another worked item is checked.
• Time pressure note: Time pressure is treated as a contributing condition, not automatically as the root cause.

Write components clearly and decide whether the target is scalar, direction, area / moment or angle. Dot product is usually the efficient route for angles and work. Cross product is the route for perpendicular direction, area and moment. Component arithmetic should be checked before interpreting the result.

Negative marking is a real exam condition, but this page does not promise a universal attempt threshold. Use method certainty and option evidence before committing to a low-confidence answer.

• Required reviewer: NDA Mathematics faculty with demonstrable subject competence.