Class 12 Mathematics Chapter 10 Vectors – Revision Notes

Vectors combines magnitude and direction into a single quantity, and this chapter covers the dot and cross products that let you test for perpendicularity, parallelism, and compute areas.

Last Updated: September 23, 2026

Basics

• |a|=√(a₁²+a₂²+a₃²); unit vector=a/|a|.
• Section formula: (mb+na)/(m+n).

Products

• Dot product: a·b=|a||b|cosθ; perpendicular ⇔ dot=0.
• Cross product: a×b=|a||b|sinθ n̂; parallel ⇔ cross=0.
• Triangle area=½|a×b|.

One-Line Summary

Vectors combine magnitude and direction; dot product relates to cosθ (perpendicularity test), cross product relates to sinθ (parallelism test and area computation).

Frequently Asked Questions

What is the difference between a scalar and a vector quantity?
A scalar quantity has only magnitude, like mass or temperature, while a vector quantity has both magnitude and direction, like displacement or force, and requires more than one number to fully describe it.

What is the geometric meaning of the dot product and cross product of two vectors?
The dot product of two vectors gives a scalar related to how much the vectors point in the same direction, useful for finding angles, while the cross product gives a vector perpendicular to both, whose magnitude relates to the area of the parallelogram they form.

Written by Satish

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