C
CraftExam
medium difficulty6 min

Vector Product, Direction and Area

The vector product of two vectors a and b is a x b = |a||b|sin θ n, where n is a unit vector perpendicular to the plane of a and b in the direction given by the right-hand rule.

In one line: Cross product points across the plane and its length gives area.

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Understand it

The cross product gives a vector. Its magnitude equals the area of the parallelogram formed by the two vectors. If a and b are parallel non-zero vectors, then a x b = 0 because sin θ = 0.

Example

i x j = k, j x k = i, k x i = j, while j x i = -k. Order matters in cross product.

Formula and rule

Key formulas: a x b = |a||b|sin θ n; |a x b| = area of parallelogram; area of triangle = (1/2)|a x b|.

Component determinant: a x b = | i j k; a1 a2 a3; b1 b2 b3 |.

Conditions: direction follows right-hand rule; zero result occurs for parallel vectors or when one vector is zero.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

  1. 1Find a x b for a = i + 2j + 3k and b = 2i - j + k.
  2. 2Using determinant, a x b = i(2(1) - 3(-1)) - j(1(1) - 3(2)) + k(1(-1) - 2(2)) = i(5) - j(-5) + k(-5) = 5i + 5j - 5k.

Method you can reuse

Write vectors in component form; set up the 3 by 3 determinant with i, j, k in the first row; expand carefully with minus sign in the j term; simplify components; use magnitude if area is required; apply right-hand rule for direction.

Sketch it in your answer

Use a diagram showing two vectors forming a parallelogram, the perpendicular direction n, and right-hand rule orientation.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What is the magnitude of a x b if |a| = 5, |b| = 4, and the angle between them is 30 degrees?

Think first, then show answer
|a x b| = |a||b|sin 30 degrees = 5 × 4 × 1/2 = 10.

Common wrong answer

Forgetting the negative sign before the j component during determinant expansion.

Don't mix this up

Students often use a x b = b x a, but cross product is anti-commutative: b x a = -(a x b).

Exam tip

When calculating by determinant, keep the signs of i, j, k components carefully, especially the middle component.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions ask for cross product, area of parallelogram or triangle, direction perpendicular to two vectors, and checking whether vectors are parallel.
  1. 1

    Enough for 1 mark

    The vector product of two vectors a and b is a x b = |a||b|sin θ n, where n is a unit vector perpendicular to the plane of a and b in the direction given by the right-hand rule.

  2. 2

    Add for 2 marks

    Example: i x j = k, j x k = i, k x i = j, while j x i = -k. Order matters in cross product.

  3. 3

    Add for 3 marks

    The cross product gives a vector.

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