Question 4 of 16intermediate💡 UnderstandLong Answer3 marks

Show that the area of the triangle contained between the vectors a and b is one half of the magnitude of a × b.

Correct Answer

The magnitude of the vector product a × b is given by |a × b| = ab sin θ, where a and b are the magnitudes of vectors a and b respectively, and θ is the angle between them.

Consider two vectors a and b forming a triangle with angle θ between them. If we take vector a as the base of the triangle, then the perpendicular height from the tip of b to a is given by b sin θ. The area of the triangle formed by these two vectors is: Area = (1/2) × base × height = (1/2) × a × (b sin θ) = (1/2) ab sin θ.

Since |a × b| = ab sin θ, we can conclude that the area of the triangle contained between vectors a and b is one half of the magnitude of a × b. Hence proved.

Exercise: EXERCISES | Q: 6.4 | (Page 34)
For More Understanding

Explanation

The question asks to prove a geometric interpretation of the vector cross product. The textbook context provides the definition that magnitude of a × b = ab sin θ. The proof connects this formula to the standard triangle area formula (1/2 × base × height). When two vectors form adjacent sides of a triangle, the area can be expressed using the cross product magnitude.

Solution Steps

  1. Step 1: State the magnitude of vector product: |a × b| = ab sin θ

  2. Step 2: Express triangle area using base a and height b sin θ: Area = (1/2) × a × b sin θ

  3. Step 3: Compare both expressions to show Area = (1/2)|a × b|

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