Vectors Problem Example 5

Description:

Problem

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

Solution

Visualizing the triangle created by the vectors a→ and b→,

Shaded region is the area enclosed by the triangle.

• In ΔABC,

  • AC = |b→|

  • AB = |a→|

• Construct the altitude, CD, in the triangle.

• sinθ = CD/AC

• CD = AC sinθ = |b→| sin θ

Area = 1/2 × base × altitude

Area = 1/2 × AB × CD

Area = 1/2 × |a→| × |b→| sinθ

Also,

|a→ × b→| = |a→| × |b→| × sinθ

Hence,

Area = 1/2 |a→ × b→|