Shadow and area
"Shadow and area" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "Vector (cross) product" in Vector Algebra. A has length 4 and b has length 3; θ is the angle between them. In the dot view, b casts a shadow on the line of a: the projection |b| cos θ, and a·b is 4 times it. In the cross views the parallelogram on a and b is shaded: its area is |a × b| = 12 sin θ. A·b = |a||b| cos θ is a number: |a| times b's shadow on a, zero at 90° and negative past it. |a × b| = |a||b| sin θ is the parallelogram's area, zero when a and b are parallel; a × b points out of the page here and b × a = −(a × b) points in. JEE Main asked 55 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
From the lesson Vector (cross) product in Vector Algebra. Change the values and watch what happens.
In JEE Main
42 shifts analysed
55 questions from Vector Algebra across the 42 JEE Main shifts Lumi has analysed (2017–2026).
The idea behind it
NCERT §10.6.3
- For non-zero, non-parallel a and b at angle θ, a × b = |a||b| sin θ n̂, where n̂ is the unit vector perpendicular to both, chosen so that a, b, n̂ form a right-handed system: turn the fingers of the right hand from a to b and the thumb points along n̂. If a or b is 0, or they are parallel, a × b = 0.
- The result is a vector. For non-zero a and b, a × b = 0 exactly when a and b are parallel (collinear); in particular a × a = 0.
- Order matters: b × a = −(a × b). For the axes, î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ, while ĵ × î = −k̂ and î × î = 0.
- The angle from the cross product: sin θ = |a × b|/(|a||b|). The cross product distributes over addition, a × (b + c) = a × b + a × c, and λ(a × b) = (λa) × b = a × (λb).
- In components, a × b is the determinant with rows (î, ĵ, k̂), (a₁, a₂, a₃), (b₁, b₂, b₃): (a₂b₃ − a₃b₂)î − (a₁b₃ − a₃b₁)ĵ + (a₁b₂ − a₂b₁)k̂.
- Worked: (2î + ĵ + 3k̂) × (3î + 5ĵ − 2k̂) = −17î + 13ĵ + 7k̂, whose length is √507.
- Worked: a vector perpendicular to both a + b and a − b, for a = î + ĵ + k̂ and b = î + 2ĵ + 3k̂, is (a + b) × (a − b) = −2î + 4ĵ − 2k̂; its length is √24 = 2√6, so a unit vector is (−î + 2ĵ − k̂)/√6, and its negative is the other one.
Questions
3 answers
- What does the "Shadow and area" simulation show?
- A has length 4 and b has length 3; θ is the angle between them. In the dot view, b casts a shadow on the line of a: the projection |b| cos θ, and a·b is 4 times it. In the cross views the parallelogram on a and b is shaded: its area is |a × b| = 12 sin θ.
- Does JEE Main ask questions from Vector Algebra?
- Yes. JEE Main asked 55 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
- Is this simulation free?
- Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Vector (cross) product" in Vector Algebra (Class 12 Maths).
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