Example 1 · level 1
Find λ so that a = −2ĵ + k̂ is perpendicular to b = î + λk̂.
Show solution
- a · b = 0 + λ = 0.
- λ = 0.
Answer: -0
CLASS 12 MATHS · VECTOR ALGEBRA
Dot product, projection and cross product
a · b = a₁b₁ + a₂b₂ + a₃b₃. Projection of a on b = (a · b)/|b|. |a × b| is the area of the parallelogram on a and b.
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Worked examples
Example 2 · level 2
Find the projection of a = î + ĵ − 4k̂ on b = 2î + 9ĵ + 6k̂.
Show solution
- a · b = −13.
- |b| = 11.
- Projection = −13/11 = −13/11.
Answer: -13/11
Example 3 · level 3
Find the area of the parallelogram with adjacent sides a = −2î − 2ĵ + k̂ and b = 2î + 3ĵ + 4k̂ (in square units).
Show solution
- a × b = −11î + 10ĵ − 2k̂.
- |a × b| = √(121 + 100 + 4) = 15.
- Area = 15 sq units.
Answer: 15