Question 5 of 16intermediate💡 UnderstandLong Answer3 marks

Show that a . (b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors a, b and c.

Correct Answer

Let us consider three vectors a, b, and c forming adjacent edges of a parallelepiped.

First, consider the vector product b × c. From the definition of vector product, the magnitude of b × c is |b||c| sin θ, where θ is the angle between b and c. This magnitude |b × c| represents the area of the parallelogram formed by vectors b and c, which forms the base of the parallelepiped. The vector b × c is perpendicular to the plane containing b and c.

Now, the scalar product a · (b × c) has magnitude |a||b × c| cos φ, where φ is the angle between a and (b × c). Since (b × c) is perpendicular to the base parallelogram, the quantity |a| cos φ represents the height of the parallelepiped (perpendicular distance from the tip of a to the base).

Therefore, |a · (b × c)| = Area of base × Height = Volume of the parallelepiped formed by vectors a, b, and c.

Exercise: EXERCISES | Q: 6.5 | (Page 34)
For More Understanding

Explanation

This question tests understanding of scalar triple product. The key insight is that |b × c| gives the area of the base parallelogram (from the definition of vector product magnitude = ab sin θ), and when we take the dot product with a, the cos φ factor extracts the perpendicular height component. The product of base area and height gives volume. The context provides the definition of vector product magnitude (ab sin θ) and its perpendicular property, which are essential for this derivation.

Solution Steps

  1. Step 1: Identify that |b × c| = |b||c| sin θ represents area of base parallelogram

  2. Step 2: Note that (b × c) is perpendicular to the plane containing b and c

  3. Step 3: Apply scalar product: |a · (b × c)| = |a||b × c| cos φ

  4. Step 4: Recognize that |a| cos φ = height of parallelepiped

  5. Step 5: Conclude: Volume = Area × Height = |b × c| × |a| cos φ = |a · (b × c)|

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