Example 18 - Class 12

Transcript

Example 18 (Introduction) Find the vector equations of the plane passing through the points R(2, 5, – 3), S(– 2, – 3, 5) and T(5, 3,– 3). Vector equation of a plane passing through three points with position vectors 𝑎﷯, 𝑏﷯, 𝑐﷯ is ( r﷯ − 𝑎﷯) . ( 𝑏﷯− 𝑎﷯)×( 𝑐﷯− 𝑎﷯)﷯ = 0 Example 18 Find the vector equations of the plane passing through the points R(2, 5, – 3), S(– 2, – 3, 5) and T(5, 3,– 3). Vector equation of a plane passing through three points with position vectors 𝑎﷯, 𝑏﷯, 𝑐﷯ is ( r﷯ − 𝒂﷯) . ( 𝒃﷯− 𝒂﷯)×( 𝒄﷯− 𝒂﷯)﷯ = 0 Now the planes passes through the points ( 𝒄﷯ − 𝒂﷯) = (5 𝑖﷯ + 3 𝑗﷯ − 3 𝑘﷯) − (2 𝑖﷯ + 5 𝑗﷯ − 3 𝑘﷯) = (5 − 2) 𝑖﷯ + (3 − 5) 𝑗﷯ + (3 − (− 3)) 𝑘﷯ = 3 𝑖﷯ − 2 𝑗﷯ + 0 𝑘﷯ = 3 𝒊﷯ − 2 𝒋﷯ Vector equation is ( r﷯ − 𝑎﷯) . ( 𝑏﷯− 𝑎﷯)×( 𝑐﷯− 𝑎﷯)﷯ = 0 ( 𝒓﷯ − (2 𝒊﷯ + 5 𝒋﷯ − 3 𝒌﷯)). −4 𝒊﷯ − 8 𝒋﷯ + 8 𝒌﷯ ﷯ × 3 𝒊﷯ − 2 𝒋﷯﷯﷯ = 0